MIME-Version: 1.0 Content-Type: multipart/related; boundary="----=_NextPart_01C48638.3C956210" This document is a Web archive file. If you are seeing this message, this means your browser or editor doesn't support Web archive files. For more information on the Web archive format, go to http://officeupdate.microsoft.com/office/webarchive.htm ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <!--[if gte mso 9]><xml> <o:DocumentProperties> <o:Author>Duncan Simester</o:Author> <o:LastAuthor>Shie Mannor</o:LastAuthor> <o:Revision>47</o:Revision> <o:TotalTime>3322</o:TotalTime> <o:Created>2004-03-09T14:02:22Z</o:Created> <o:LastSaved>2004-08-20T02:02:36Z</o:LastSaved> <o:Words>447</o:Words> <o:PresentationFormat>On-screen Show</o:PresentationFormat> <o:Company>MIT Sloan</o:Company> <o:Bytes>327166</o:Bytes> <o:Paragraphs>138</o:Paragraphs> <o:Slides>21</o:Slides> <o:Version>10.2625</o:Version> </o:DocumentProperties> <o:OfficeDocumentSettings> <o:PixelsPerInch>80</o:PixelsPerInch> </o:OfficeDocumentSettings> </xml><![endif]--> <link rel=3DPresentation-XML href=3D"BiasVarianceICML1_files/pres.xml"> <meta name=3DDescription content=3D"8/19/2004: Bias and Variance in Value Function Estimates Shie = Mannor (MIT McGill) Joint work with: Duncan Simester (MIT) Peng S= un (Duke) John N. 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href=3D"javascript:parent.CollapseOutline()"><img src=3D"v3_outline_collapse.gif" border=3D0 alt=3D"Collapse Outline"></a><a href=3D"javascript:parent.ExpandOutline()"><img src=3D"v3_outline_expand.gi= f" border=3D0 alt=3D"Expand Outline"></a></center> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_outline_collapsed.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_outline_collapsed.ht= m" id=3D"webarch_temp_base_tag"> <![endif]><script> <!-- if( window.name !=3D "PPTOtl" ) parent.location.href =3D "../BiasVarianceICML1.htm"; //--> </script> <meta name=3DRobots content=3DNoIndex> </head> <body bgcolor=3Dblack alink=3Dwhite link=3Dwhite vlink=3Dwhite text=3Dwhite> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0002.htm');">Bias and Variance <br> in Value Function Estimates<br> <br> Shie Mannor (MIT<span style=3D'mso-spacerun:yes'>       </span>McGill)<br> Joint work with:<span style=3D'mso-spacerun:yes'>  </span>Duncan Simester (MIT)<br> <span style=3D'mso-spacerun:yes'>  </span>Peng Sun (Duke)<br> John N. Tsitsiklis (MIT)</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0051.htm');">Estimating Value Fu= nction from Data</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0052.htm');">Estimating Value Functions</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0053.htm');">Parametric vs. Inte= rnal Variance</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0025.htm');">Illustration<br> The Catalog Circulation Problem</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0015.htm');">Illustration</a></b= ></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0018.htm');">Estimating Value Functions</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0026.htm');">Slide 8</a></b></fo= nt></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0021.htm');">Bias and Variance</= a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0027.htm');">Slide 10</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0022.htm');">Analysis: Fixed Pol= icy</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0054.htm');">Analysis</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0028.htm');">Validation</a></b><= /font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0035.htm');">Recall Catalog Data= </a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0036.htm');">Slide 15</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0048.htm');">Control: Improving = the Policy</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0038.htm');">Take home message</= a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0039.htm');">Future Research</a>= </b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0050.htm');">The Control Problem= </a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0045.htm');">Slide 20</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0046.htm');">Slide 21</a></b></f= ont></p> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_outline_expanded.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_outline_expand= ed.htm" id=3D"webarch_temp_base_tag"> <![endif]><script> <!-- if( window.name !=3D "PPTOtl" ) parent.location.href =3D "../BiasVarianceICML1.htm"; //--> </script> <meta name=3DRobots content=3DNoIndex> </head> <body bgcolor=3Dblack alink=3Dwhite link=3Dwhite vlink=3Dwhite text=3Dwhite> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0002.htm');">Bias and Variance <br> in Value Function Estimates<br> <br> Shie Mannor (MIT<span style=3D'mso-spacerun:yes'>       </span>McGill)<br> Joint work with:<span style=3D'mso-spacerun:yes'>  </span>Duncan Simester (MIT)<br> <span style=3D'mso-spacerun:yes'>  </span>Peng Sun (Duke)<br> John N. Tsitsiklis (MIT)</a></b></font></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0051.htm');">Estimating Value Fu= nction from Data</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>So you have collected your da= ta</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Estimate rewards and transiti= ons</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>For each action in each state= </font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Estimate value function for a= fixed policy</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Improve policy</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>How good is your estimate? (c= onfidence)</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0052.htm');">Estimating Value Functions</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Value function:</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Rearranging, we obtain:</font= ></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Errors in R and P lead to err= ors in V.</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0053.htm');">Parametric vs. Inte= rnal Variance</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Internal:</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Parametric:</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Different entities, used for = different things, different probabilistic setup</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0025.htm');">Illustration<br> The Catalog Circulation Problem</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Women’s clothing retailer</fo= nt></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>1.7 million customers</font><= /td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Entire purchase history</font= ></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>6 years of mailing history</f= ont></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0015.htm');">Illustration</a></b= ></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>64 states:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Recency, Frequency and Moneta= ry Value</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Quartile (4<sup>3</sup>)</fon= t></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>250 Sub-samples:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>6,880 and 657,000 observation= s in each</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>10,000 / reward and 1,400 / t= ransition</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>“True” model</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>All 1.7 million customers</fo= nt></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0018.htm');">Estimating Value Functions</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>64 states (RFM)</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Estimate rewards and transiti= ons</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>250 parameter estimates</font= ></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Policy evaluation</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>250 value function estimates<= /font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Compare with “True” value fun= ction</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0026.htm');">Slide 8</a></b></fo= nt></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0021.htm');">Bias and Variance</= a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Bias:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>True mean:<span style=3D'mso-spacerun:yes'>  </span>$79.01</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Mean across sub-samples:<span style=3D'mso-spacerun:yes'>  </span>$79.20</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Standard deviation:<span style=3D'mso-spacerun:yes'>  </span>$4.95</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Variance:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>95% confidence interval:<span style=3D'mso-spacerun:yes'>  </span>$69.11 to $88.91</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Robustness:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Vary size of sub-samples</fon= t></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Vary discount rate</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0027.htm');">Slide 10</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0022.htm');">Analysis: Fixed Pol= icy</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Derive 1st and 2nd moments of= V</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Write:</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Assume:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Use sample values of the model parameters</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>2nd order approximation</font= ></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0054.htm');">Analysis</a></b></f= ont> <table> <tr> <td width=3D5 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Two approaches:</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Classical – Derive expression= s using true P and R and then substitute them.</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Bayesian – Assume comfortable= priors (Dirichlet and Gaussian)</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1><font face=3DArial size=3D2>Same expressions unless there= is a meaningful prior</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D1></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0028.htm');">Validation</a></b><= /font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Check accuracy of confidence = intervals</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Evaluate assumptions</font></= td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Use of sample model parameter= s</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>2<sup>nd</sup> order approxim= ation</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Control problem</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Policy no longer fixed</font>= </td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0035.htm');">Recall Catalog Data= </a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Sub-samples:</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Equally sized sub-samples</fo= nt></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>“True” model: all 1.7 million= customers</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Confidence intervals:</font><= /td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Confidence bound for each sub= -sample</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>% times True value falls with= in bounds</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Repeat 10 times (redraw sub-s= amples)</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0036.htm');">Slide 15</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0048.htm');">Control: Improving = the Policy</a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Compare current policy vs. “o= ptimal” policy</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Optimal policy contains anoth= er source of bias</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Optimization favors positive = errors</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Errors in rewards and transit= ions are no longer unbiased</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Bias in rewards and transitio= ns leads to bias in value function estimates</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0038.htm');">Take home message</= a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Small error in parameter esti= mate may lead to big error in value function</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Illustrate magnitudes</font><= /td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Variance is large</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Bias is smaller but larger fo= r the control problem</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Closed form expressions</font= ></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Bias and variance</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Confidence bounds</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Compare current and “optimal”= policies</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0039.htm');">Future Research</a>= </b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Use variance estimates to gui= de exploration</font></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Improve control algorithms</f= ont></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Transitions derived from mode= l free estimates</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Parametric approaches (model = belongs to a small family)</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1><font face=3DArial size=3D2>Nested estimation</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0050.htm');">The Control Problem= </a></b></font> <table> <tr> <td width=3D5 nowrap></td> <td width=3D20 nowrap></td> <td width=3D"100%"></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Cross-validate to remove bias= </font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Derive optimal policy using a calibration sample</font></td> </tr> <tr> <td colspan=3D2></td> <td colspan=3D1></td> </tr> <tr> <td colspan=3D1></td> <td colspan=3D2><font face=3DArial size=3D2>Evaluate policy using validat= ion sample</font></td> </tr> </table> </p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0045.htm');">Slide 20</a></b></f= ont></p> <p><font face=3DArial size=3D2><b><a href=3D"javascript:parent.GoToSld('v3_slide0046.htm');">Slide 21</a></b></f= ont></p> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_blank_notes.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File 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/title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0002.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Bias and Variance in Value Function Estimates Shie = Mannor (MIT McGill) Joint work with: Duncan Simester (MIT) Peng S= un (Duke) John N. Tsitsiklis (MIT)"> <link rel=3Dnext href=3D"v3_slide0051.htm"> </head> <body background=3D"v3_slide0002_background.gif" bgcolor=3Dwhite alink=3D"#= 333399" link=3D"#009999" vlink=3D"#99CC00" text=3Dblack topmargin=3D0 leftmargin=3D= 0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D83 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D310 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D475 height=3D69 colspan=3D3></td> </tr> <tr> <td width=3D83 height=3D229></td> <td width=3D310 height=3D229> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D3 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D8 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D3 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D41 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D2 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D197 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D2 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D40 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D3 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D8 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D55 height=3D24 colspan=3D4></td> <td width=3D201 height=3D24 colspan=3D3 align=3Dcenter><font face=3DAri= al size=3D4><b>Bias and Variance</b></font><br> </td> <td width=3D54 height=3D24 colspan=3D4></td> </tr> <tr> <td width=3D310 height=3D14 colspan=3D11></td> </tr> <tr> <td width=3D310 height=3D24 colspan=3D11 align=3Dcenter><font face=3DAr= ial size=3D4><b>in Value Function Estimates</b></font><br> </td> </tr> <tr> <td width=3D310 height=3D56 colspan=3D11></td> </tr> <tr> <td width=3D3 height=3D24></td> <td width=3D304 height=3D24 colspan=3D9 align=3Dcenter><font face=3DAri= al size=3D4>Shie Mannor (MIT<span style=3D'mso-spacerun:yes'>       </span>McGill)</font= ><br> </td> <td width=3D3 height=3D24></td> </tr> <tr> <td width=3D310 height=3D11 colspan=3D11></td> </tr> <tr> <td width=3D11 height=3D19 colspan=3D2></td> <td width=3D288 height=3D19 colspan=3D7 align=3Dcenter><font face=3DAri= al size=3D1>Joint work with:<span style=3D'mso-spacerun:yes'>  </span></font><font face= =3DArial size=3D2>Duncan Simester (MIT)</font><br> </td> <td width=3D11 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D310 height=3D10 colspan=3D11></td> </tr> <tr> <td width=3D57 height=3D19 colspan=3D5></td> <td width=3D197 height=3D19 align=3Dcenter><font face=3DArial size=3D2>= <span style=3D'mso-tab-count:1'>   </span><span style=3D'mso-spacerun:yes'>  </span>Peng Sun (Duke)</font><br> </td> <td width=3D56 height=3D19 colspan=3D5></td> </tr> <tr> <td width=3D310 height=3D9 colspan=3D11></td> </tr> <tr> <td width=3D14 height=3D19 colspan=3D3></td> <td width=3D282 height=3D19 colspan=3D5 align=3Dcenter><font face=3DAri= al size=3D2><span style=3D'mso-tab-count:2'>     </span>John N. Tsitsiklis (MIT)</font><b= r> </td> <td width=3D14 height=3D19 colspan=3D3></td> </tr> </table> </td> <td width=3D82 height=3D229></td> </tr> <tr> <td width=3D475 height=3D58 colspan=3D3></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0002_background.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlh3AFlAXcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAADb AWQBgQAAAAAAAP///wECAwL/lI+py+0Po5y02ouz3rz7D4biSJbmiabqyrbuC8fyTNf2jef6zvf+ DwwKh8Si8YhMKpfMpvMJjUqn1Kr1is1qt9yu9wsOi8fksvmMTqvX7Lb7DY/L5/S6/Y7P6/f8vv8P GCg4SFhoeIiYqLjI2Oj4CBkpOUlZaXmJmam5ydnp+QkaKjpKWmp6ipqqusra6voKGys7S1tre4ub q7vL2+v7CxwsPExcbHyMnKy8zNzs/AwdLT1NXW19jZ2tvc3d7f0NHi4+Tl5ufo6err7O3u7+Dh8v P09fb3+Pn6+/z9/v/w8woMCBBAsaPIgwocKFDBs6fAgxosSJFCtavIgxo8aN/xw7evwIMqTIkSRL mjyJMqXKlSxbunwJM6bMmTRr2ryJM6fOnTx7+vwJNKjQoUSLGj2KNKnSpUybOn0KNarUqVSrWr2K NavWrVy7ev0KNqzYsWTLmj2LNq3atWzbun0LN67cuXTr2r2LN6/evXz7+v0LOLDgwYQLGz6MOLHi xYwbO34MObLkyZQrW76MObPmzZw7e/4MOrTo0aRLmz6NOrXq1axbu34NO7ZsdgECdLA9W0Lt2hx2 536wO7jw4cSL4/69wLjy5byRN2i+Abrz6dSrW7+OPbv27dy7e/8OPrz48eTLmz+PPr369ezbu38P P778+fTr27+PP7/+/fz7+4H/D2CAAg5IYIEGHohgggouyGCDDj4IYYQSTkhhhRZeiGGGGm7IYYce fghiiCKOSGKJJp6IYooqrshiiy6+CGOMMs5IY4023ohjjjruyGOPPv4IZJBCDklkkUYeiWSSSi7J ZJNOPglllFJOSWWVVl6JZZZabslll15+CWaYYo5JSwEAOw== ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0051.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0051.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Function fr= om Data"> <link rel=3Dnext href=3D"v3_slide0052.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D35 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D25 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D353 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D35 height=3D24></td> <td width=3D407 height=3D24 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D407 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Estimating Value Function from Data</b></font><br> </td> </tr> </table> </td> <td width=3D33 height=3D24></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D60 height=3D227 colspan=3D2></td> <td width=3D353 height=3D227> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D30 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D18 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D77 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D136 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D6 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D64 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D277 height=3D22 colspan=3D6><font face=3DArial size=3D3>So = you have collected your data</font><br> </td> <td width=3D76 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D353 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D283 height=3D22 colspan=3D7><font face=3DArial size=3D3>Est= imate rewards and transitions</font><br> </td> <td width=3D70 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D353 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D30 height=3D19></td> <td width=3D18 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D214 height=3D19 colspan=3D3><font face=3DArial size=3D2>For= each action in each state</font><br> </td> <td width=3D91 height=3D19 colspan=3D4></td> </tr> <tr> <td width=3D353 height=3D27 colspan=3D9></td> </tr> <tr> <td width=3D347 height=3D22 colspan=3D8><font face=3DArial size=3D3>Est= imate value function for a fixed policy</font><br> </td> <td width=3D6 height=3D22></td> </tr> <tr> <td width=3D353 height=3D32 colspan=3D9></td> </tr> <tr> <td width=3D126 height=3D22 colspan=3D4><font face=3DArial size=3D3>Imp= rove policy</font><br> </td> <td width=3D227 height=3D22 colspan=3D5></td> </tr> <tr> <td width=3D353 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D353 height=3D22 colspan=3D9><font face=3DArial size=3D3>How= good is your estimate? (confidence)</font><br> </td> </tr> </table> </td> <td width=3D62 height=3D227 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D42 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0052.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0052.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Functions"> <link rel=3Dnext href=3D"v3_slide0053.htm"> </head> <body background=3D"v3_slide0052_background.gif" bgcolor=3Dwhite alink=3D"#= 333399" link=3D"#009999" vlink=3D"#99CC00" text=3Dblack topmargin=3D0 leftmargin=3D= 0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D56 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D213 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D92 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D85 height=3D24 colspan=3D2></td> <td width=3D305 height=3D24 colspan=3D2> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D305 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Estimating Value Functions</b></font><br> </td> </tr> </table> </td> <td width=3D85 height=3D24></td> </tr> <tr> <td width=3D475 height=3D29 colspan=3D5></td> </tr> <tr> <td width=3D29 height=3D223></td> <td width=3D269 height=3D223 colspan=3D2> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D111 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D67 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D111 height=3D19><font face=3DArial size=3D2>Value function:= </font><br> </td> <td width=3D158 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D269 height=3D72 colspan=3D3></td> </tr> <tr> <td width=3D178 height=3D19 colspan=3D2><font face=3DArial size=3D2>Rea= rranging, we obtain:</font><br> </td> <td width=3D91 height=3D19></td> </tr> <tr> <td width=3D269 height=3D95 colspan=3D3></td> </tr> <tr> <td width=3D269 height=3D19 colspan=3D3><font face=3DArial size=3D2>Err= ors in <i>R</i> and <i>P</i> lead to errors in <i>V</i>.</font><br> </td> </tr> </table> </td> <td width=3D177 height=3D223 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D47 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0052_background.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdh2wFkAXcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAA2wFkAYAAAAD/ //8C/4yPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2Py+f0uv2Oz+v3/L7/DxgoOEhYaHiImKi4yNjo +AgZKTlJWWl5iZmpucnZ6fkJGio6SlpqeoqaqrrK2ur6ChsrO0tba3uLm6u7y9vr+wscLDxMXGx8 jJysvMzc7PwMHS09TV1tfY2drb3N3e39DR4uPk5ebn6Onq6+zt7u/g4fLz9PX29/j5+vv8/f7/8P MKDAgQQLGjyIMKHChQwbOnwIMaLEiRQrWryIMaPGjf8cO3r8CDKkyJEkS5o8iTKlypUsW7p8CTOm 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{behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0053.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Parametric vs. Internal Vari= ance"> <link rel=3Dnext href=3D"v3_slide0025.htm"> </head> <body background=3D"v3_slide0053_background.gif" bgcolor=3Dwhite alink=3D"#= 333399" link=3D"#009999" vlink=3D"#99CC00" text=3Dblack topmargin=3D0 leftmargin=3D= 0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D33 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D353 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D50 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D62 height=3D24 colspan=3D2></td> <td width=3D353 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D353 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Parametric vs. Internal Variance</b></font><br> </td> </tr> </table> </td> <td width=3D60 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D29 height=3D203></td> <td width=3D436 height=3D203 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D17 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D54 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D74 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D71 height=3D22 colspan=3D2><font face=3DArial size=3D3>Inte= rnal:</font><br> </td> <td width=3D365 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D436 height=3D58 colspan=3D5></td> </tr> <tr> <td width=3D100 height=3D22 colspan=3D3><font face=3DArial color=3D"#33= 3399" size=3D3>Parametric</font><font face=3DArial color=3Dblack size=3D3>:</font><br> </td> <td width=3D336 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D436 height=3D57 colspan=3D5></td> </tr> <tr> <td width=3D436 height=3D22 colspan=3D5><font face=3DArial size=3D3>Dif= ferent entities, used for different things, different</font><br> </td> </tr> <tr> <td width=3D436 height=3D1 colspan=3D5></td> </tr> <tr> <td width=3D17 height=3D22></td> <td width=3D157 height=3D22 colspan=3D3><font face=3DArial size=3D3>pro= babilistic setup</font><br> </td> <td width=3D262 height=3D22></td> </tr> </table> </td> <td width=3D10 height=3D203></td> </tr> <tr> <td width=3D475 height=3D66 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 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rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0025.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Illustration The Catalog Circulation Problem"> <link rel=3Dnext href=3D"v3_slide0015.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D58 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D71 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D219 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D70 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D58 height=3D50></td> <td width=3D360 height=3D50 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D120 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D119 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> </tr> <tr> <td width=3D120 height=3D24></td> <td width=3D119 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Illustration</b></font><br> </td> <td width=3D121 height=3D24></td> </tr> <tr> <td width=3D360 height=3D2 colspan=3D3></td> </tr> <tr> <td width=3D360 height=3D24 colspan=3D3 align=3Dcenter><font face=3DAri= al size=3D4><b>The Catalog Circulation Problem</b></font><br> </td> </tr> </table> </td> <td width=3D57 height=3D50></td> </tr> <tr> <td width=3D475 height=3D46 colspan=3D5></td> </tr> <tr> <td width=3D129 height=3D121 colspan=3D2></td> <td width=3D219 height=3D121> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D183 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D17 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D16 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D219 height=3D22 colspan=3D4><font face=3DArial size=3D3>Wom= en’s clothing retailer</font><br> </td> </tr> <tr> <td width=3D219 height=3D11 colspan=3D4></td> </tr> <tr> <td width=3D183 height=3D22><font face=3DArial size=3D3>1.7 million cus= tomers</font><br> </td> <td width=3D36 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D219 height=3D11 colspan=3D4></td> </tr> <tr> <td width=3D200 height=3D22 colspan=3D2><font face=3DArial size=3D3>Ent= ire purchase history</font><br> </td> <td width=3D19 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D219 height=3D12 colspan=3D4></td> </tr> <tr> <td width=3D216 height=3D22 colspan=3D3><font face=3DArial size=3D3>6 y= ears of mailing history</font><br> </td> <td width=3D3 height=3D22></td> </tr> </table> </td> <td width=3D127 height=3D121 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D106 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0015.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Illustration"> <link rel=3Dnext href=3D"v3_slide0018.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D68 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D110 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D119 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D117 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D178 height=3D24 colspan=3D2></td> <td width=3D119 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D119 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Illustration</b></font><br> </td> </tr> </table> </td> <td width=3D178 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D26 colspan=3D5></td> </tr> <tr> <td width=3D68 height=3D238></td> <td width=3D346 height=3D238 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D45 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D27 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D17 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D61 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D96 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D21 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D85 height=3D22 colspan=3D4><font face=3DArial size=3D3>64 s= tates:</font><br> </td> <td width=3D261 height=3D22 colspan=3D7></td> </tr> <tr> <td width=3D346 height=3D3 colspan=3D11></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3D"#333399" size=3D= 2>–</font></td> <td width=3D307 height=3D19 colspan=3D9><font face=3DArial color=3D"#33= 3399" size=3D2>R</font><font face=3DArial color=3Dblack size=3D2>ecency, </font><font face=3DArial color=3D"#333399" size=3D2>F</font><font face=3DArial color=3Dblack siz= e=3D2>requency and </font><font face=3DArial color=3D"#333399" size=3D2>M</font><font face=3DArial color=3Dblack size=3D2>onetary Value</font><br> </td> </tr> <tr> <td width=3D346 height=3D4 colspan=3D11></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D90 height=3D20 colspan=3D4 rowspan=3D2><font face=3DArial s= ize=3D2>Quartile (4<sup>3</sup>)</font><br> </td> <td width=3D217 height=3D19 colspan=3D5></td> </tr> <tr> <td width=3D39 height=3D1 colspan=3D2></td> <td width=3D217 height=3D1 colspan=3D5></td> </tr> <tr> <td width=3D346 height=3D27 colspan=3D11></td> </tr> <tr> <td width=3D158 height=3D22 colspan=3D7><font face=3DArial size=3D3>250= Sub-samples:</font><br> </td> <td width=3D188 height=3D22 colspan=3D4></td> </tr> <tr> <td width=3D346 height=3D3 colspan=3D11></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D297 height=3D19 colspan=3D8><font face=3DArial size=3D2>6,8= 80 and 657,000 observations in each</font><br> </td> <td width=3D10 height=3D19></td> </tr> <tr> <td width=3D346 height=3D4 colspan=3D11></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D276 height=3D19 colspan=3D7><font face=3DArial size=3D2>10,= 000 / reward and 1,400 / transition</font><br> </td> <td width=3D31 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D346 height=3D31 colspan=3D11></td> </tr> <tr> <td width=3D112 height=3D22 colspan=3D5><font face=3DArial size=3D3>“Tr= ue” model</font><br> </td> <td width=3D234 height=3D22 colspan=3D6></td> </tr> <tr> <td width=3D346 height=3D4 colspan=3D11></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D180 height=3D19 colspan=3D6><font face=3DArial size=3D2>All= 1.7 million customers</font><br> </td> <td width=3D127 height=3D19 colspan=3D3></td> </tr> </table> </td> <td width=3D61 height=3D238></td> </tr> <tr> <td width=3D475 height=3D35 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0018.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Functions"> <link rel=3Dnext href=3D"v3_slide0026.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D85 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D3 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></td> <td width=3D302 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D6 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></td> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D85 height=3D24></td> <td width=3D305 height=3D24 colspan=3D2> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D305 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Estimating Value Functions</b></font><br> </td> </tr> </table> </td> <td width=3D85 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D88 height=3D192 colspan=3D2></td> <td width=3D308 height=3D192 colspan=3D2> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D30 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D18 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D91 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D7 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D86 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D27 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D23 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D140 height=3D22 colspan=3D4><font face=3DArial size=3D3>64 = states (RFM)</font><br> </td> <td width=3D169 height=3D22 colspan=3D5></td> </tr> <tr> <td width=3D309 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D283 height=3D22 colspan=3D8><font face=3DArial size=3D3>Est= imate rewards and transitions</font><br> </td> <td width=3D26 height=3D22></td> </tr> <tr> <td width=3D309 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D30 height=3D19></td> <td width=3D18 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D185 height=3D19 colspan=3D4><font face=3DArial size=3D2>250= parameter estimates</font><br> </td> <td width=3D76 height=3D19 colspan=3D3></td> </tr> <tr> <td width=3D309 height=3D27 colspan=3D9></td> </tr> <tr> <td width=3D147 height=3D22 colspan=3D5><font face=3DArial size=3D3>Pol= icy evaluation</font><br> </td> <td width=3D162 height=3D22 colspan=3D4></td> </tr> <tr> <td width=3D309 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D30 height=3D19></td> <td width=3D18 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D212 height=3D19 colspan=3D5><font face=3DArial size=3D2>250= value function estimates</font><br> </td> <td width=3D49 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D309 height=3D3 colspan=3D9></td> </tr> <tr> <td width=3D30 height=3D19></td> <td width=3D18 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D261 height=3D19 colspan=3D7><font face=3DArial size=3D2>Com= pare with “True” value function</font><br> </td> </tr> </table> </td> <td width=3D79 height=3D192></td> </tr> <tr> <td width=3D475 height=3D77 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0026.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004"> <link rel=3Dnext href=3D"v3_slide0021.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D41 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D360 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D19 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D41 height=3D24></td> <td width=3D394 height=3D24 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D394 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Value Function:<span style=3D'mso-spacerun:yes'>  </span>True vs. Estimated</= b></font><br> </td> </tr> </table> </td> <td width=3D40 height=3D24></td> </tr> <tr> <td width=3D475 height=3D10 colspan=3D5></td> </tr> <tr> <td width=3D56 height=3D270 colspan=3D2></td> <td width=3D360 height=3D270><img src=3D"v3_slide0026_image001.jpg" borde= r=3D0 width=3D360 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href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0021.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Bias and Variance"> <link rel=3Dnext href=3D"v3_slide0027.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D76 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D62 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D201 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D92 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D138 height=3D24 colspan=3D2></td> <td width=3D201 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D201 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Bias and Variance</b></font><br> </td> </tr> </table> </td> <td width=3D136 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D28 colspan=3D5></td> </tr> <tr> <td width=3D76 height=3D237></td> <td width=3D355 height=3D237 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D4 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D39 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D26 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D65 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D11 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D41 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D8 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D65 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D44 height=3D22 colspan=3D4><font face=3DArial size=3D3>Bias= :</font><br> </td> <td width=3D311 height=3D22 colspan=3D8></td> </tr> <tr> <td width=3D355 height=3D2 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D146 height=3D19 colspan=3D6><font face=3DArial size=3D2>Tru= e mean:<span style=3D'mso-spacerun:yes'>  </span>$79.01</font><br> </td> <td width=3D170 height=3D19 colspan=3D4></td> </tr> <tr> <td width=3D355 height=3D1 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D260 height=3D19 colspan=3D9><font face=3DArial size=3D2>Mea= n across sub-samples:<span style=3D'mso-spacerun:yes'>  </span>$79.20</font><br> </td> <td width=3D56 height=3D19></td> </tr> <tr> <td width=3D355 height=3D2 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D195 height=3D19 colspan=3D8><font face=3DArial size=3D2>Sta= ndard deviation:<span style=3D'mso-spacerun:yes'>  </span>$4.95</font><br> </td> <td width=3D121 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D355 height=3D24 colspan=3D12></td> </tr> <tr> <td width=3D83 height=3D22 colspan=3D5><font face=3DArial size=3D3>Vari= ance:</font><br> </td> <td width=3D272 height=3D22 colspan=3D7></td> </tr> <tr> <td width=3D355 height=3D1 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D316 height=3D19 colspan=3D10><font face=3DArial size=3D2>95= % confidence interval:<span style=3D'mso-spacerun:yes'>  </span>$69.11 to $88.91</fo= nt><br> </td> </tr> <tr> <td width=3D355 height=3D24 colspan=3D12></td> </tr> <tr> <td width=3D109 height=3D22 colspan=3D6><font face=3DArial size=3D3>Rob= ustness:</font><br> </td> <td width=3D246 height=3D22 colspan=3D6></td> </tr> <tr> <td width=3D355 height=3D1 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D187 height=3D19 colspan=3D7><font face=3DArial size=3D2>Var= y size of sub-samples</font><br> </td> <td width=3D129 height=3D19 colspan=3D3></td> </tr> <tr> <td width=3D355 height=3D2 colspan=3D12></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D135 height=3D19 colspan=3D5><font face=3DArial size=3D2>Var= y discount rate</font><br> </td> <td width=3D181 height=3D19 colspan=3D5></td> </tr> </table> </td> <td width=3D44 height=3D237></td> </tr> <tr> <td width=3D475 height=3D34 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0027.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0027.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004"> <link rel=3Dnext href=3D"v3_slide0022.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D63 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D40 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D269 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D43 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D38 colspan=3D5></td> </tr> <tr> <td width=3D103 height=3D24 colspan=3D2></td> <td width=3D269 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D269 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Varying the Sample Size</b></font><br> </td> </tr> </table> </td> <td width=3D103 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D10 colspan=3D5></td> </tr> <tr> <td width=3D63 height=3D264></td> <td width=3D352 height=3D264 colspan=3D3><img src=3D"v3_slide0027_image00= 2.jpg" border=3D0 width=3D352 height=3D264></td> <td width=3D60 height=3D264></td> </tr> <tr> <td width=3D475 height=3D20 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0027_image002.jpg 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</style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0022.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Analysis: Fixed Policy"> <link rel=3Dnext href=3D"v3_slide0054.htm"> </head> <body background=3D"v3_slide0022_background.gif" bgcolor=3Dwhite alink=3D"#= 333399" link=3D"#009999" vlink=3D"#99CC00" text=3Dblack topmargin=3D0 leftmargin=3D= 0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D68 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D47 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D246 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D72 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D115 height=3D24 colspan=3D2></td> <td width=3D246 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D246 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Analysis: Fixed Policy</b></font><br> </td> </tr> </table> </td> <td width=3D114 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D28 colspan=3D5></td> </tr> <tr> <td width=3D68 height=3D210></td> <td width=3D365 height=3D210 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D12 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D144 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D69 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D289 height=3D22 colspan=3D7><font face=3DArial size=3D3>Der= ive 1st and 2nd moments of <i>V</i></font><br> </td> <td width=3D76 height=3D22></td> </tr> <tr> <td width=3D365 height=3D51 colspan=3D8></td> </tr> <tr> <td width=3D52 height=3D22 colspan=3D4><font face=3DArial size=3D3>Writ= e:</font><br> </td> <td width=3D313 height=3D22 colspan=3D4></td> </tr> <tr> <td width=3D365 height=3D51 colspan=3D8></td> </tr> <tr> <td width=3D76 height=3D22 colspan=3D5><font face=3DArial size=3D3>Assu= me:</font><br> </td> <td width=3D289 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D365 height=3D2 colspan=3D8></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D326 height=3D19 colspan=3D6><font face=3DArial size=3D2>Use= sample values of the model parameters</font><br> </td> </tr> <tr> <td width=3D365 height=3D2 colspan=3D8></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D181 height=3D19 colspan=3D4><font face=3DArial size=3D2>2nd= order approximation</font><br> </td> <td width=3D145 height=3D19 colspan=3D2></td> </tr> </table> </td> <td width=3D42 height=3D210></td> </tr> <tr> <td width=3D475 height=3D61 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0022_background.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODdh2wFkAXcAACH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACwAAAAA2wFkAYAAAAD/ //8C/4yPqcvtD6OctNqLs968+w+G4kiW5omm6sq27gvH8kzX9o3n+s73/g8MCofEovGITCqXzKbz CY1Kp9Sq9YrNarfcrvcLDovH5LL5jE6r1+y2+w2Py+f0uv2Oz+v3/L7/DxgoOEhYaHiImKi4yNjo +AgZKTlJWWl5iZmpucnZ6fkJGio6SlpqeoqaqrrK2ur6ChsrO0tba3uLm6u7y9vr+wscLDxMXGx8 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<![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0054.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Analysis"> <link rel=3Dnext href=3D"v3_slide0028.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D29 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D161 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D96 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D175 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D190 height=3D24 colspan=3D2></td> <td width=3D96 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D96 height=3D24 align=3Dcenter><font face=3DArial size=3D4><= b>Analysis</b></font><br> </td> </tr> </table> </td> <td width=3D189 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D29 height=3D221></td> <td width=3D432 height=3D221 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D17 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D40 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D87 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D6 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D86 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D172 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D8 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D150 height=3D22 colspan=3D4><font face=3DArial size=3D3>Two= approaches:</font><br> </td> <td width=3D282 height=3D22 colspan=3D4></td> </tr> <tr> <td width=3D432 height=3D4 colspan=3D8></td> </tr> <tr> <td width=3D432 height=3D22 colspan=3D8><font face=3DArial size=3D3>Cla= ssical – Derive expressions using true P and R</font><br> </td> </tr> <tr> <td width=3D17 height=3D22></td> <td width=3D219 height=3D22 colspan=3D4><font face=3DArial size=3D3>and= then substitute them.</font><br> </td> <td width=3D196 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D432 height=3D32 colspan=3D8></td> </tr> <tr> <td width=3D416 height=3D22 colspan=3D7><font face=3DArial size=3D3>Bay= esian – Assume comfortable priors (Dirichlet</font><br> </td> <td width=3D16 height=3D22></td> </tr> <tr> <td width=3D17 height=3D22></td> <td width=3D127 height=3D22 colspan=3D2><font face=3DArial size=3D3>and= Gaussian)</font><br> </td> <td width=3D288 height=3D22 colspan=3D5></td> </tr> <tr> <td width=3D432 height=3D31 colspan=3D8></td> </tr> <tr> <td width=3D408 height=3D22 colspan=3D6><font face=3DArial size=3D3>Sam= e expressions unless there is a meaningful</font><br> </td> <td width=3D24 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D17 height=3D22></td> <td width=3D40 height=3D22><font face=3DArial size=3D3>prior</font><br> </td> <td width=3D375 height=3D22 colspan=3D6></td> </tr> </table> </td> <td width=3D14 height=3D221></td> </tr> <tr> <td width=3D475 height=3D48 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0028.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0028.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Validation"> <link rel=3Dnext href=3D"v3_slide0035.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D68 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D114 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D112 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D112 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D182 height=3D24 colspan=3D2></td> <td width=3D112 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D112 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Validation</b></font><br> </td> </tr> </table> </td> <td width=3D181 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D68 height=3D192></td> <td width=3D338 height=3D192 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D90 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D52 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D16 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D16 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D71 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D338 height=3D22 colspan=3D9><font face=3DArial size=3D3>Che= ck accuracy of confidence intervals</font><br> </td> </tr> <tr> <td width=3D338 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D191 height=3D22 colspan=3D5><font face=3DArial size=3D3>Eva= luate assumptions</font><br> </td> <td width=3D147 height=3D22 colspan=3D4></td> </tr> <tr> <td width=3D338 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D24 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D246 height=3D19 colspan=3D6><font face=3DArial size=3D2>Use= of sample model parameters</font><br> </td> <td width=3D44 height=3D19></td> </tr> <tr> <td width=3D338 height=3D3 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D24 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D175 height=3D20 colspan=3D5 rowspan=3D2><font face=3DArial = size=3D2>2<sup>nd</sup> order approximation</font><br> </td> <td width=3D115 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D48 height=3D1 colspan=3D2></td> <td width=3D115 height=3D1 colspan=3D2></td> </tr> <tr> <td width=3D338 height=3D27 colspan=3D9></td> </tr> <tr> <td width=3D139 height=3D22 colspan=3D4><font face=3DArial size=3D3>Con= trol problem</font><br> </td> <td width=3D199 height=3D22 colspan=3D5></td> </tr> <tr> <td width=3D338 height=3D3 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D24 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D159 height=3D19 colspan=3D4><font face=3DArial size=3D2>Pol= icy no longer fixed</font><br> </td> <td width=3D131 height=3D19 colspan=3D3></td> </tr> </table> </td> <td width=3D69 height=3D192></td> </tr> <tr> <td width=3D475 height=3D77 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0035.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0035.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Recall Catalog Data"> <link rel=3Dnext href=3D"v3_slide0036.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D72 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D57 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D217 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D56 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D129 height=3D24 colspan=3D2></td> <td width=3D217 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D217 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Recall Catalog Data</b></font><br> </td> </tr> </table> </td> <td width=3D129 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D72 height=3D185></td> <td width=3D330 height=3D185 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D80 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D62 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D55 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D85 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D4 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D120 height=3D22 colspan=3D4><font face=3DArial size=3D3>Sub= -samples:</font><br> </td> <td width=3D210 height=3D22 colspan=3D6></td> </tr> <tr> <td width=3D330 height=3D3 colspan=3D10></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D198 height=3D19 colspan=3D4><font face=3DArial size=3D2>Equ= ally sized sub-samples</font><br> </td> <td width=3D93 height=3D19 colspan=3D4></td> </tr> <tr> <td width=3D330 height=3D4 colspan=3D10></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D283 height=3D19 colspan=3D5><font face=3DArial size=3D2>“Tr= ue” model: all 1.7 million customers</font><br> </td> <td width=3D8 height=3D19 colspan=3D3></td> </tr> <tr> <td width=3D330 height=3D28 colspan=3D10></td> </tr> <tr> <td width=3D182 height=3D22 colspan=3D5><font face=3DArial size=3D3>Con= fidence intervals:</font><br> </td> <td width=3D148 height=3D22 colspan=3D5></td> </tr> <tr> <td width=3D330 height=3D3 colspan=3D10></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D291 height=3D19 colspan=3D8><font face=3DArial size=3D2>Con= fidence bound for each sub-sample</font><br> </td> </tr> <tr> <td width=3D330 height=3D4 colspan=3D10></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D284 height=3D19 colspan=3D6><font face=3DArial size=3D2>% t= imes <i>True</i> value falls within bounds</font><br> </td> <td width=3D7 height=3D19 colspan=3D2></td> </tr> <tr> <td width=3D330 height=3D4 colspan=3D10></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D288 height=3D19 colspan=3D7><font face=3DArial size=3D2>Rep= eat 10 times (redraw sub-samples)</font><br> </td> <td width=3D3 height=3D19></td> </tr> </table> </td> <td width=3D73 height=3D185></td> </tr> <tr> <td width=3D475 height=3D84 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape 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Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0048.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Control: Improving the Polic= y"> <link rel=3Dnext href=3D"v3_slide0038.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D48 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D27 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D326 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D45 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D46 colspan=3D5></td> </tr> <tr> <td width=3D75 height=3D24 colspan=3D2></td> <td width=3D326 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D326 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Control: Improving the Policy</b></font><br> </td> </tr> </table> </td> <td width=3D74 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D43 colspan=3D5></td> </tr> <tr> <td width=3D48 height=3D181></td> <td width=3D398 height=3D181 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D68 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D110 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D73 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D77 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D14 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D368 height=3D22 colspan=3D7><font face=3DArial size=3D3>Com= pare current policy vs. “optimal” policy</font><br> </td> <td width=3D30 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D398 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D398 height=3D22 colspan=3D9><font face=3DArial size=3D3>Opt= imal policy contains another source of bias</font><br> </td> </tr> <tr> <td width=3D398 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D252 height=3D19 colspan=3D4><font face=3DArial size=3D2>Opt= imization favors positive errors</font><br> </td> <td width=3D107 height=3D19 colspan=3D3></td> </tr> <tr> <td width=3D398 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D343 height=3D19 colspan=3D6><font face=3DArial size=3D2>Err= ors in rewards and transitions are no longer</font><br> </td> <td width=3D16 height=3D19></td> </tr> <tr> <td width=3D39 height=3D18 colspan=3D2></td> <td width=3D69 height=3D18 colspan=3D2><font face=3DArial size=3D2>unbi= ased</font><br> </td> <td width=3D290 height=3D18 colspan=3D5></td> </tr> <tr> <td width=3D398 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D343 height=3D19 colspan=3D6><font face=3DArial size=3D2>Bia= s in rewards and transitions leads to bias in</font><br> </td> <td width=3D16 height=3D19></td> </tr> <tr> <td width=3D39 height=3D19 colspan=3D2></td> <td width=3D179 height=3D19 colspan=3D3><font face=3DArial size=3D2>val= ue function estimates</font><br> </td> <td width=3D180 height=3D19 colspan=3D4></td> </tr> </table> </td> <td width=3D29 height=3D181></td> </tr> <tr> <td width=3D475 height=3D62 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0038.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Take home message"> <link rel=3Dnext href=3D"v3_slide0039.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D92 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D32 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D228 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D81 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D124 height=3D24 colspan=3D2></td> <td width=3D228 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D228 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Take home message</b></font><br> </td> </tr> </table> </td> <td width=3D123 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D29 colspan=3D5></td> </tr> <tr> <td width=3D92 height=3D220></td> <td width=3D341 height=3D220 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D18 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D6 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D14 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D2 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D102 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D7 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D9 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D10 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D32 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D44 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D33 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D56 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D341 height=3D19 colspan=3D13><font face=3DArial size=3D2><b= >Small error in parameter estimate may lead</b></font><br> </td> </tr> <tr> <td width=3D18 height=3D19></td> <td width=3D226 height=3D19 colspan=3D9><font face=3DArial size=3D2><b>= to big error in value function</b></font><br> </td> <td width=3D97 height=3D19 colspan=3D3></td> </tr> <tr> <td width=3D341 height=3D27 colspan=3D13></td> </tr> <tr> <td width=3D168 height=3D19 colspan=3D8><font face=3DArial size=3D2><b>= Illustrate magnitudes</b></font><br> </td> <td width=3D173 height=3D19 colspan=3D5></td> </tr> <tr> <td width=3D341 height=3D3 colspan=3D13></td> </tr> <tr> <td width=3D24 height=3D16 colspan=3D2></td> <td width=3D14 height=3D16><font face=3DArial color=3Dblack size=3D1>–<= /font></td> <td width=3D104 height=3D16 colspan=3D2><font face=3DArial size=3D1>Var= iance is large</font><br> </td> <td width=3D199 height=3D16 colspan=3D8></td> </tr> <tr> <td width=3D341 height=3D3 colspan=3D13></td> </tr> <tr> <td width=3D24 height=3D16 colspan=3D2></td> <td width=3D14 height=3D16><font face=3DArial color=3Dblack size=3D1>–<= /font></td> <td width=3D295 height=3D16 colspan=3D9><font face=3DArial size=3D1>Bia= s is smaller but larger for the control problem</font><br> </td> <td width=3D8 height=3D16></td> </tr> <tr> <td width=3D341 height=3D22 colspan=3D13></td> </tr> <tr> <td width=3D200 height=3D19 colspan=3D9><font face=3DArial size=3D2><b>= Closed form expressions</b></font><br> </td> <td width=3D141 height=3D19 colspan=3D4></td> </tr> <tr> <td width=3D341 height=3D3 colspan=3D13></td> </tr> <tr> <td width=3D24 height=3D16 colspan=3D2></td> <td width=3D14 height=3D16><font face=3DArial color=3Dblack size=3D1>–<= /font></td> <td width=3D111 height=3D16 colspan=3D3><font face=3DArial size=3D1>Bia= s and variance</font><br> </td> <td width=3D192 height=3D16 colspan=3D7></td> </tr> <tr> <td width=3D341 height=3D3 colspan=3D13></td> </tr> <tr> <td width=3D24 height=3D16 colspan=3D2></td> <td width=3D14 height=3D16><font face=3DArial color=3Dblack size=3D1>–<= /font></td> <td width=3D120 height=3D16 colspan=3D4><font face=3DArial size=3D1>Con= fidence bounds</font><br> </td> <td width=3D183 height=3D16 colspan=3D6></td> </tr> <tr> <td width=3D341 height=3D3 colspan=3D13></td> </tr> <tr> <td width=3D24 height=3D16 colspan=3D2></td> <td width=3D14 height=3D16><font face=3DArial color=3Dblack size=3D1>–<= /font></td> <td width=3D239 height=3D16 colspan=3D8><font face=3DArial size=3D1>Com= pare current and “optimal” policies</font><br> </td> <td width=3D64 height=3D16 colspan=3D2></td> </tr> </table> </td> <td width=3D42 height=3D220></td> </tr> <tr> <td width=3D475 height=3D50 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0039.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0039.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Future Research"> <link rel=3Dnext href=3D"v3_slide0050.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D52 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D94 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D184 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D128 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D146 height=3D24 colspan=3D2></td> <td width=3D184 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D184 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>Future Research</b></font><br> </td> </tr> </table> </td> <td width=3D145 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D30 colspan=3D5></td> </tr> <tr> <td width=3D52 height=3D193></td> <td width=3D406 height=3D193 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D15 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D49 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D84 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D57 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D148 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D17 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> </tr> <tr> <td width=3D378 height=3D22 colspan=3D7><font face=3DArial size=3D3>Use= variance estimates to guide exploration</font><br> </td> <td width=3D28 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D406 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D230 height=3D22 colspan=3D6><font face=3DArial size=3D3>Imp= rove control algorithms</font><br> </td> <td width=3D176 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D406 height=3D31 colspan=3D9></td> </tr> <tr> <td width=3D395 height=3D22 colspan=3D8><font face=3DArial size=3D3>Tra= nsitions derived from model free estimates</font><br> </td> <td width=3D11 height=3D22></td> </tr> <tr> <td width=3D406 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D367 height=3D19 colspan=3D7><font face=3DArial size=3D2>Par= ametric approaches (model belongs to a small</font><br> </td> </tr> <tr> <td width=3D39 height=3D19 colspan=3D2></td> <td width=3D50 height=3D19 colspan=3D2><font face=3DArial size=3D2>fami= ly)</font><br> </td> <td width=3D317 height=3D19 colspan=3D5></td> </tr> <tr> <td width=3D406 height=3D4 colspan=3D9></td> </tr> <tr> <td width=3D24 height=3D19></td> <td width=3D15 height=3D19><font face=3DArial color=3Dblack size=3D2>–<= /font></td> <td width=3D134 height=3D19 colspan=3D3><font face=3DArial size=3D2>Nes= ted estimation</font><br> </td> <td width=3D233 height=3D19 colspan=3D4></td> </tr> </table> </td> <td width=3D17 height=3D193></td> </tr> <tr> <td width=3D475 height=3D76 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0050.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0050.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: The Control Problem"> <link rel=3Dnext href=3D"v3_slide0045.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D48 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D75 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D230 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D65 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D46 colspan=3D5></td> </tr> <tr> <td width=3D123 height=3D24 colspan=3D2></td> <td width=3D230 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D230 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>The Control Problem</b></font><br> </td> </tr> </table> </td> <td width=3D122 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D47 colspan=3D5></td> </tr> <tr> <td width=3D48 height=3D142></td> <td width=3D370 height=3D142 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D28 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D1 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D63 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0><= /td> <td width=3D166 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> <td width=3D109 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0>= </td> </tr> <tr> <td width=3D258 height=3D22 colspan=3D4><font face=3DArial size=3D3>Cro= ss-validate to remove bias</font><br> </td> <td width=3D112 height=3D22 colspan=3D2></td> </tr> <tr> <td width=3D370 height=3D27 colspan=3D6></td> </tr> <tr> <td width=3D28 height=3D22><font face=3DArial color=3Dblack size=3D3>1.= </font></td> <td width=3D342 height=3D22 colspan=3D5><font face=3DArial size=3D3>Der= ive optimal policy using a calibration</font><br> </td> </tr> <tr> <td width=3D28 height=3D22></td> <td width=3D64 height=3D22 colspan=3D2><font face=3DArial size=3D3>samp= le</font><br> </td> <td width=3D278 height=3D22 colspan=3D3></td> </tr> <tr> <td width=3D370 height=3D28 colspan=3D6></td> </tr> <tr> <td width=3D28 height=3D22><font face=3DArial color=3Dblack size=3D3>2.= </font></td> <td width=3D339 height=3D22 colspan=3D4><font face=3DArial size=3D3>Eva= luate policy using validation sample</font><br> </td> <td width=3D3 height=3D22></td> </tr> </table> </td> <td width=3D57 height=3D142></td> </tr> <tr> <td width=3D475 height=3D97 colspan=3D5></td> </tr> </table> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/v3_slide0045.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="windows-1252" <html> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dwindows-125= 2"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0045.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004"> <link rel=3Dnext href=3D"v3_slide0046.htm"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D64 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D24 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D300 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D26 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D88 height=3D24 colspan=3D2></td> <td width=3D300 height=3D24> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td width=3D300 height=3D24 align=3Dcenter><font face=3DArial size=3D4>= <b>The Control Problem:<span style=3D'mso-spacerun:yes'>  </span>Bias</b></font><br> </td> </tr> </table> </td> <td width=3D87 height=3D24 colspan=3D2></td> </tr> <tr> <td width=3D475 height=3D14 colspan=3D5></td> </tr> <tr> <td width=3D64 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and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\v3_slide0046.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004"> </head> <body bgcolor=3Dwhite alink=3D"#333399" link=3D"#009999" vlink=3D"#99CC00" = text=3Dblack topmargin=3D0 leftmargin=3D0> <table border=3D0 valign=3Dtop width=3D475 height=3D356 cellpadding=3D0 cel= lspacing=3D0> <tr> <td width=3D44 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D23 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> <td width=3D348 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></= td> <td width=3D18 height=3D0 nowrap><img src=3D"v3_space.gif" border=3D0></t= d> </tr> <tr> <td width=3D475 height=3D33 colspan=3D5></td> </tr> <tr> <td width=3D44 height=3D24></td> <td width=3D389 height=3D24 colspan=3D3> <table border=3D0 valign=3Dtop cellpadding=3D0 cellspacing=3D0> <tr> <td 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xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0002.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Bias and Variance in Value Function Estimates Shie = Mannor (MIT McGill) Joint work with: Duncan Simester (MIT) Peng S= un (Duke) John N. Tsitsiklis (MIT)"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style> .CT {color:black;} </style> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"3"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s3074" type=3D"= #_x0000_m1026" style=3D'position:absolute;left:54pt;top:66pt;width:612pt;height:408pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"centerTitle"/></v:shape><v:shapetype id=3D"_x0000_= t13" coordsize=3D"21600,21600" o:spt=3D"13" adj=3D"16200,5400" path=3D"m@0,l@0= @1,0@1,0@2@0@2@0,21600,21600,10800xe"> <v:stroke joinstyle=3D"miter"/> <v:formulas> <v:f eqn=3D"val #0"/> <v:f eqn=3D"val #1"/> <v:f eqn=3D"sum height 0 #1"/> <v:f eqn=3D"sum 10800 0 #1"/> <v:f eqn=3D"sum width 0 #0"/> <v:f eqn=3D"prod @4 @3 10800"/> <v:f eqn=3D"sum width 0 @5"/> </v:formulas> <v:path o:connecttype=3D"custom" o:connectlocs=3D"@0,0;0,10800;@0,21600;2= 1600,10800" o:connectangles=3D"270,180,90,0" textboxrect=3D"0,@1,@6,@2"/> <v:handles> <v:h position=3D"#0,#1" xrange=3D"0,21600" yrange=3D"0,10800"/> </v:handles> </v:shapetype><v:shape id=3D"_x0000_s3075" type=3D"#_x0000_t13" style=3D'p= osition:absolute; left:420pt;top:4in;width:42pt;height:30pt;mso-wrap-style:none; v-text-anchor:middle' filled=3D"f" fillcolor=3D"#bbe0e3 [4]" stroked=3D"f" strokecolor=3D"black [1]"> <v:fill color2=3D"white [0]"/> <v:shadow color=3D"gray [2]"/> </v:shape><v:shape id=3D"_x0000_s3076" type=3D"#_x0000_t13" style=3D'posit= ion:absolute; left:3in;top:12pt;width:30pt;height:30pt;mso-wrap-style:none;v-text-ancho= r:middle' filled=3D"f" fillcolor=3D"#bbe0e3 [4]" stroked=3D"f" strokecolor=3D"black= [1]"> <v:fill color2=3D"white [0]"/> <v:shadow color=3D"gray [2]"/> </v:shape> <div v:shape=3D"_x0000_s3074" class=3DCT style=3D'mso-line-spacing:"150 0 = 0"'><span style=3D'position:absolute;top:19.25%;left:8.42%;width:83.14%;height:6.75%= '><span style=3D'mso-bidi-font-family:Arial;color:black;mso-color-index:1'>Bias and Variance <br> </span></span><span style=3D'position:absolute;top:30.0%;left:8.42%;width= :83.14%; height:6.75%'><span style=3D'mso-bidi-font-family:Arial;color:black;mso-co= lor-index: 1'>in Value Function Estimates</span><span style=3D'font-size:113%'><br> </span></span><span style=3D'position:absolute;top:41.0%;left:8.42%;width= :83.14%; height:7.75%'><span style=3D'font-size:113%'><br> </span></span><span style=3D'position:absolute;top:52.75%;left:8.42%; width:83.14%;height:6.75%'><span style=3D'font-weight:normal'>Shie Mannor (MIT<span style=3D'mso-spacerun:yes'>       </span>McGill)<br> </span></span><span style=3D'position:absolute;top:62.5%;left:8.42%;width= :83.14%; height:5.25%'><span style=3D'font-size:63%;font-weight:normal'>Joint work with:<span style=3D'mso-spacerun:yes'>  </span></span><span style=3D'font-size:75%;font-weight:normal'>Duncan Simester (MIT)<br> </span></span><span style=3D'position:absolute;top:70.5%;left:8.42%;width= :83.14%; height:5.25%'><span style=3D'font-size:75%;font-weight:normal'><span style=3D'mso-tab-count:1;width:12.02%'> </span><span style=3D'mso-spacerun:yes'>  </span>Peng Sun (Duke)<br> </span></span><span style=3D'position:absolute;top:78.5%;left:8.42%;width= :83.14%; height:5.25%'><span style=3D'font-size:75%;font-weight:normal'><span style=3D'mso-tab-count:2;width:24.05%'> </span>John N. Tsitsiklis (MIT)</s= pan></span></div> <v:line id=3D"_x0000_s3077" style=3D'position:absolute;v-text-anchor:middl= e' from=3D"414pt,300pt" to=3D"462pt,300pt" strokecolor=3D"black [1]" strokew= eight=3D"3pt"> <v:stroke startarrow=3D"diamond" endarrow=3D"block"/> <v:shadow color=3D"gray [2]"/> </v:line><![if !vml]><img border=3D0 v:shapes=3D"_x0000_s3077" src=3D"slide0002_image006.gif" style=3D'position:absolute;top:54.5%;left:5= 6.74%; width:8.05%;height:2.75%'><![endif]></p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0002_image006.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhLAAMAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAMAAwAm AAYAgAAAAAAAAAIdBBKmy+3XEHq0SiWz3rzn44UiB07VyXxYgraGaRQAOx== ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0051.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0051.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Function fr= om Data"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"62"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s63490" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s63491" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:84pt;top:126pt;width:588pt;height:5in'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s63490" class=3DT style=3D'position:absolute;top:9.= 25%; left:3.37%;width:93.44%;height:6.75%'>Estimating Value Function from Data<= /div> <div v:shape=3D"_x0000_s63491"> <div class=3DB2 style=3D'mso-margin-left-alt:1103;mso-text-indent-alt:863'= ></div> <div class=3DB3 style=3D'mso-margin-left-alt:1415;mso-text-indent-alt:1175= '></div> <div class=3DB4 style=3D'mso-margin-left-alt:1727;mso-text-indent-alt:1487= '></div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:24.2= 5%; left:12.73%;width:79.77%;height:6.0%'>So you have collected your data<span style=3D'font-size:114%;display:none'> </span></div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:31.7= 5%; left:12.73%;width:79.77%;height:6.0%'>Estimate rewards and transitions = ;</div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:39.0%;left:18.91%;width:73.59%;height:5.25%'><span style=3D'position:absolute;top:0%;left:5.08%;width:94.91%;height:100.0%'><= span style=3D'mso-special-format:bullet;position:absolute;left:-5.36%'>–<= /span>For each action in each state </span></div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:45.5%;left:18.91%;width:73.59%;height:5.25%'><span style=3D'mso-special-format:nobullet;display:none'>–</span> </di= v> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:52.0= %; left:12.73%;width:83.52%;height:6.0%'>Estimate value function for a fixed policy </div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:59.5= %; left:12.73%;width:79.77%;height:6.0%'> </div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:67.0= %; left:12.73%;width:79.77%;height:6.0%'>Improve policy </div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:74.5= %; left:12.73%;width:79.77%;height:6.0%'> </div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:82.0= %; left:12.73%;width:85.01%;height:6.0%'>How good is your estimate? (confiden= ce)</div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0052.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <link rel=3DOLE-Object-Data href=3Doledata.mso> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0052.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Functions"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"63"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s64514" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s64515" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:36pt;top:126pt;width:480pt;height:356.375= pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1" size=3D"half"/></v:shape> <div v:shape=3D"_x0000_s64514" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Estimating Value Functions</div> <div v:shape=3D"_x0000_s64515" class=3DHB> <div style=3D'position:absolute;top:24.25%;left:5.99%;width:64.79%;height:= 5.25%'>Value function: </div> <div style=3D'position:absolute;top:30.75%;left:5.99%;width:64.79%;height:= 5.25%'> </div> <div style=3D'position:absolute;top:37.0%;left:5.99%;width:64.79%;height:5= .25%'> </div> <div style=3D'position:absolute;top:43.5%;left:5.99%;width:64.79%;height:5= .25%'> </div> <div style=3D'position:absolute;top:49.75%;left:5.99%;width:64.79%;height:= 5.25%'>Rearranging, we obtain: </div> <div style=3D'position:absolute;top:56.25%;left:5.99%;width:64.79%;height:= 5.25%'> </div> <div style=3D'position:absolute;top:62.5%;left:5.99%;width:64.79%;height:5= .25%'> </div> <div style=3D'position:absolute;top:69.0%;left:5.99%;width:64.79%;height:5= .25%'> </div> <div style=3D'position:absolute;top:75.5%;left:5.99%;width:64.79%;height:5= .25%'> </div> <div style=3D'position:absolute;top:81.75%;left:5.99%;width:64.79%;height:= 5.25%'>Errors in <i>R</i> and <i>P</i> lead to errors in <i>V</i>. </div> </div> <p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s6= 4516" type=3D"#_x0000_m1027" style=3D'position:absolute;left:178pt;top:174.5pt;= width:278pt; height:50.5pt' o:spt=3D"75" o:preferrelative=3D"t" path=3D"m@4@5l@4@11@9@= 11@9@5xe" filled=3D"f" fillcolor=3D"white" stroked=3D"f" strokecolor=3D"black"> <v:fill opacity=3D"1" color2=3D"white" o:opacity2=3D"1" recolor=3D"f" rot= ate=3D"f" o:detectmouseclick=3D"f"/> <v:stroke joinstyle=3D"miter" o:forcedash=3D"f"/> <v:imagedata src=3D"slide0052_image007.wmz" o:title=3D""/> <v:shadow on=3D"f" 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Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0053.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Parametric vs. Internal Vari= ance"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"65"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s66562" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s66563" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:36pt;top:126pt;width:9in;height:356.375pt= '> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s66562" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Parametric vs. Internal Variance<span lang=3DHE style=3D'mso-special-format:lastCR;display:none'> </span></d= iv> <div v:shape=3D"_x0000_s66563" class=3DB> <div style=3D'position:absolute;top:24.25%;left:5.99%;width:88.2%;height:6= .0%'>Internal: </div> <div style=3D'position:absolute;top:31.75%;left:5.99%;width:88.2%;height:6= .0%'> </div> <div style=3D'position:absolute;top:39.25%;left:5.99%;width:88.2%;height:6= .0%'> </div> <div style=3D'position:absolute;top:46.75%;left:5.99%;width:88.2%;height:6= .0%'><span style=3D'color:#333399;mso-color-index:5'>Parametric</span>: </div> <div style=3D'position:absolute;top:54.25%;left:5.99%;width:88.2%;height:6= .0%'> </div> <div style=3D'position:absolute;top:61.75%;left:5.99%;width:88.2%;height:6= .0%'> </div> <span style=3D'position:absolute;top:69.25%;left:5.99%;width:104.86%;heigh= t: 6.0%'>Different entities, used for different things, different </span><span style=3D'position:absolute;top:75.5%;left:9.73%;width:84.45%;height:6.0%'>= probabilistic setup<span lang=3DHE style=3D'mso-special-format:lastCR;display:none'> = ;</span></span></div> <v:shapetype id=3D"_x0000_t75" coordsize=3D"21600,21600" o:spt=3D"75" o:preferrelative=3D"t" path=3D"m@4@5l@4@11@9@11@9@5xe" filled=3D"f" strok= ed=3D"f"> <v:stroke joinstyle=3D"miter"/> <v:formulas> <v:f eqn=3D"if lineDrawn pixelLineWidth 0"/> <v:f eqn=3D"sum @0 1 0"/> <v:f eqn=3D"sum 0 0 @1"/> <v:f eqn=3D"prod @2 1 2"/> <v:f eqn=3D"prod @3 21600 pixelWidth"/> <v:f eqn=3D"prod @3 21600 pixelHeight"/> <v:f eqn=3D"sum @0 0 1"/> <v:f eqn=3D"prod @6 1 2"/> <v:f eqn=3D"prod @7 21600 pixelWidth"/> <v:f eqn=3D"sum @8 21600 0"/> <v:f eqn=3D"prod @7 21600 pixelHeight"/> <v:f eqn=3D"sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" 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style=3D'position:absolute;left:188.125pt;top:181.875pt;width:393.875pt; height:310.125pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s31750" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:42pt;width:9in;height:90pt; v-text-anchor:middle' o:spt=3D"1" path=3D"m,l,21600r21600,l21600,xe" fill= ed=3D"f" fillcolor=3D"#bbe0e3 [4]" stroked=3D"f" strokecolor=3D"black [1]"> <v:fill color2=3D"white [0]" o:detectmouseclick=3D"f"/> <v:stroke joinstyle=3D"miter" o:forcedash=3D"f"/> <v:shadow color=3D"gray [2]"/> <v:path gradientshapeok=3D"t" o:connecttype=3D"rect"/> <o:lock v:ext=3D"edit" text=3D"f" grouping=3D"t"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s31747" class=3DB> <div style=3D'mso-line-spacing:"130 20 0";position:absolute;top:36.25%; left:27.15%;width:52.8%;height:6.0%'>Women’s clothing retailer = </div> <div style=3D'mso-line-spacing:"130 20 0";position:absolute;top:45.5%; left:27.15%;width:52.8%;height:6.0%'>1.7 million customers </div> <div style=3D'mso-line-spacing:"130 20 0";position:absolute;top:55.0%; left:27.15%;width:52.8%;height:6.0%'>Entire purchase history </div> <div style=3D'mso-line-spacing:"130 20 0";position:absolute;top:64.25%; left:27.15%;width:52.8%;height:6.0%'>6 years of mailing history</div> </div> <div v:shape=3D"_x0000_s31750" class=3DT><span style=3D'position:absolute; top:9.5%;left:5.99%;width:88.2%;height:6.75%'>Illustration<br> </span><span style=3D'position:absolute;top:16.5%;left:5.99%;width:88.2%; height:6.75%'>The Catalog Circulation Problem</span></div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0015.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0015.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Illustration"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"18"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s18434" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s18435" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:96pt;top:120pt;width:588pt;height:396pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s18434" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Illustration</div> <div v:shape=3D"_x0000_s18435"> <div class=3DB style=3D'position:absolute;top:23.25%;left:14.41%;width:79.= 77%; height:6.0%'>64 states: </div> <div class=3DB1 style=3D'position:absolute;top:30.5%;left:19.28%;width:76.= 96%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.13%;width:95.= 86%; height:100.0%'><span style=3D'color:#333399;mso-color-index:5'><span class= =3DB1B style=3D'position:absolute;left:-4.31%'>–</span></span><span style=3D'color:#333399;mso-color-index:5'>R</span>ecency, <span style=3D'color:#333399;mso-color-index:5'>F</span>requency and <span style=3D'color:#333399;mso-color-index:5'>M</span>onetary Value </span= ></div> <div class=3DB1 style=3D'position:absolute;top:36.75%;left:19.28%;width:74= .9%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.25%;width:95.= 75%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.43%'>&= #8211;</span>Quartile (4<span style=3D'font-size:67%;position:relative;top:-.45em;mso-text-raise= :30%'>3</span>) </span></div> <div class=3DB1 style=3D'position:absolute;top:43.25%;left:19.28%;width:74= .9%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.25%;width:95.= 75%; height:100.0%'><span style=3D'visibility:hidden'><span class=3DB1B style=3D'position:absolute;left:-4.43%'>–</span></span> </span><= /div> <div class=3DB style=3D'position:absolute;top:50.0%;left:14.41%;width:79.7= 7%; height:6.0%'>250 Sub-samples: </div> <div class=3DB1 style=3D'position:absolute;top:57.0%;left:19.28%;width:74.= 9%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.25%;width:95.= 75%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.43%'>&= #8211;</span>6,880 and 657,000 observations in each </span></div> <div class=3DB1 style=3D'position:absolute;top:63.5%;left:19.28%;width:74.= 9%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.25%;width:95.= 75%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.43%'>&= #8211;</span>10,000 / reward and 1,400 / transition </span></div> <div class=3DB style=3D'position:absolute;top:70.25%;left:14.41%;width:79.= 77%; height:6.0%'> </div> <div class=3DB style=3D'position:absolute;top:77.5%;left:14.41%;width:79.7= 7%; height:6.0%'>“True” model </div> <div class=3DB1 style=3D'position:absolute;top:84.75%;left:19.28%;width:74= .9%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.25%;width:95.= 75%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.43%'>&= #8211;</span>All 1.7 million customers</span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0018.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0018.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Estimating Value Functions"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld 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href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s21506" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s21507" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:126pt;top:126pt;width:522pt;height:5in'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s21506" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Estimating Value Functions</div> <div v:shape=3D"_x0000_s21507"> <div class=3DB2 style=3D'mso-margin-left-alt:1103;mso-text-indent-alt:863'= ></div> <div class=3DB3 style=3D'mso-margin-left-alt:1415;mso-text-indent-alt:1175= '></div> <div class=3DB4 style=3D'mso-margin-left-alt:1727;mso-text-indent-alt:1487= '></div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:24.2= 5%; left:18.53%;width:70.59%;height:6.0%'>64 states (RFM) </div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:31.75%;left:24.71%;width:64.41%;height:6.0%'><span style=3D'mso-special-format:nobullet;display:none'>–</span><span style=3D'font-size:117%;display:none'> </span></div> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:39.2= 5%; left:18.53%;width:70.59%;height:6.0%'>Estimate rewards and transitions = ;</div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:46.5%;left:24.71%;width:64.41%;height:5.25%'><span style=3D'position:absolute;top:0%;left:5.81%;width:94.18%;height:100.0%'><= span style=3D'mso-special-format:bullet;position:absolute;left:-6.17%'>–<= /span>250 parameter estimates </span></div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:53.0%;left:24.71%;width:64.41%;height:5.25%'><span style=3D'mso-special-format:nobullet;display:none'>–</span> </di= v> <div class=3DB style=3D'mso-margin-left-alt:291;position:absolute;top:59.5= %; left:18.53%;width:70.59%;height:6.0%'>Policy evaluation </div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:66.75%;left:24.71%;width:64.41%;height:5.25%'><span style=3D'position:absolute;top:0%;left:5.81%;width:94.18%;height:100.0%'><= span style=3D'mso-special-format:bullet;position:absolute;left:-6.17%'>–<= /span>250 value function estimates </span></div> <div class=3DB1 style=3D'mso-margin-left-alt:578;mso-text-indent-alt:363; position:absolute;top:73.25%;left:24.71%;width:66.47%;height:5.25%'><span style=3D'position:absolute;top:0%;left:5.63%;width:94.36%;height:100.0%'><= span style=3D'mso-special-format:bullet;position:absolute;left:-5.97%'>–<= /span>Compare with “True” value function</span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0026.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview 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<![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0021.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Bias and Variance"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"27"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); 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v:shape=3D"_x0000_s27650" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Bias and Variance</div> <div v:shape=3D"_x0000_s27651"> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:2= 3.75%; left:15.91%;width:74.9%;height:6.0%'>Bias: </div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 30.5%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.77%'>–</span>True mean:<span style=3D'mso-spacerun:yes'>  </span>$79.01 </span></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 36.25%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.77%'>–</span>Mean across sub-samples:<span style=3D'mso-spacerun:yes'>  </span>$79.20 </s= pan></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 42.25%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.77%'>–</span>Standard deviation:<sp= an style=3D'mso-spacerun:yes'>  </span>$4.95 </span></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 48.0%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'visibility:hi= dden'><span style=3D'mso-special-format:bullet;position:absolute;left:-4.77%'>–<= /span></span> </span></div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:5= 4.0%; left:15.91%;width:74.9%;height:6.0%'>Variance: </div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 60.75%; left:20.97%;width:79.21%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.01%;width:95.98%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.18%'>–</span>95% confidence interval:<span style=3D'mso-spacerun:yes'>  </span>$69.11 to $88.91&#= 13;</span></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 66.75%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'visibility:hi= dden'><span style=3D'mso-special-format:bullet;position:absolute;left:-4.77%'>–<= /span></span> </span></div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:7= 2.75%; left:15.91%;width:74.9%;height:6.0%'>Robustness: </div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 79.25%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.77%'>–</span>Vary size of sub-samples </span></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 85.25%; left:20.97%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:4.55%;width:95.44%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-4.77%'>–</span>Vary discount rate</sp= an></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0027.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; 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<v:fill color2=3D"white [0]" o:detectmouseclick=3D"f"/> <v:stroke joinstyle=3D"miter" o:forcedash=3D"f"/> <v:shadow color=3D"gray [2]"/> <v:path gradientshapeok=3D"t" o:connecttype=3D"rect"/> <o:lock v:ext=3D"edit" text=3D"f" grouping=3D"t"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s28674" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape> <div v:shape=3D"_x0000_s28682"> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:2= 3.75%; left:14.41%;width:74.9%;height:6.0%'>Derive 1st and 2nd moments of <i>V= 3;</i></div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:3= 0.75%; left:14.41%;width:74.9%;height:6.0%'> </div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:3= 7.5%; left:14.41%;width:74.9%;height:6.0%'> </div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:4= 4.5%; left:14.41%;width:74.9%;height:6.0%'>Write: </div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:5= 1.25%; left:14.41%;width:74.9%;height:6.0%'> </div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:5= 8.0%; left:14.41%;width:74.9%;height:6.0%'> </div> <div class=3DB style=3D'mso-line-spacing:"90 20 0";position:absolute;top:6= 5.0%; left:14.41%;width:74.9%;height:6.0%'>Assume: </div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 0";position:absolute;top:= 71.5%; left:19.28%;width:81.64%;height:5.25%'><span style=3D'position:absolute; top:0%;left:3.89%;width:96.1%;height:100.0%'><span style=3D'mso-special-fo= rmat: bullet;position:absolute;left:-4.05%'>–</span>Use sample values of t= he model parameters </span></div> <div class=3DB1 style=3D'mso-line-spacing:"90 20 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class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s67586" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s67587" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:36pt;top:126pt;width:9in;height:356.375pt= '> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s67586" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Analysis<span lang=3DHE style=3D'mso-= special-format: lastCR;display:none'> </span></div> <div v:shape=3D"_x0000_s67587" class=3DB> <div style=3D'position:absolute;top:24.25%;left:5.99%;width:88.2%;height:6= .0%'>Two approaches: </div> <span style=3D'position:absolute;top:31.75%;left:5.99%;width:103.93%;heigh= t: 6.0%'>Classical – Derive expressions using true P and R </span><span style=3D'position:absolute;top:38.0%;left:9.73%;width:84.45%;height:6.0%'>= and then substitute them. </span> <div style=3D'position:absolute;top:45.5%;left:5.99%;width:88.2%;height:6.= 0%'> </div> <span style=3D'position:absolute;top:53.0%;left:5.99%;width:100.0%;height:= 6.0%'>Bayesian – Assume comfortable priors (Dirichlet </span><span style=3D'positio= n: absolute;top:59.25%;left:9.73%;width:84.45%;height:6.0%'>and Gaussian) = ;</span> <div style=3D'position:absolute;top:66.75%;left:5.99%;width:88.2%;height:6= .0%'> </div> <span style=3D'position:absolute;top:74.25%;left:5.99%;width:98.12%;height= :6.0%'>Same expressions unless there is a meaningful </span><span style=3D'position:ab= solute; top:80.5%;left:9.73%;width:84.45%;height:6.0%'>prior </span> <div style=3D'position:absolute;top:88.0%;left:5.99%;width:88.2%;height:6.= 0%'><span style=3D'mso-special-format:lastCR;display:none'> </span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0028.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0028.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Validation"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"37"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s37890" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s37891" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:96pt;top:126pt;width:552pt;height:356.375= pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s37890" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Validation</div> <div v:shape=3D"_x0000_s37891"> <div class=3DB2 style=3D'mso-margin-left-alt:864'></div> <div class=3DB3 style=3D'mso-margin-left-alt:1152'></div> <div class=3DB4 style=3D'mso-margin-left-alt:1440'></div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:24.2= 5%; left:14.41%;width:81.27%;height:6.0%'>Check accuracy of confidence intervals </div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:31.7= 5%; left:14.41%;width:74.9%;height:6.0%'> </div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:39.2= 5%; left:14.41%;width:74.9%;height:6.0%'>Evaluate assumptions </div> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:46.= 5%; left:19.28%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:7.23%;width:92.76%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-7.8%'>–</span>Use of sample model parameters </span></div> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:53.= 0%; left:19.28%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:7.23%;width:92.76%;height:100.0%'><span style=3D'mso-special-f= ormat: bullet;position:absolute;left:-7.8%'>–</span>2<span style=3D'font-si= ze: 67%;position:relative;top:-.45em;mso-text-raise:30%'>nd</span><span style=3D'mso-spacerun:yes'> </span>order approximation </span></d= iv> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:59.= 25%; left:19.28%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:7.23%;width:92.76%;height:100.0%'><span style=3D'visibility:hi= dden'><span style=3D'mso-special-format:bullet;position:absolute;left:-7.8%'>–</= span></span> </span></div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:66.0= %; left:14.41%;width:74.9%;height:6.0%'>Control problem </div> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:73.= 25%; left:19.28%;width:69.85%;height:5.25%'><span style=3D'position:absolute; top:0%;left:7.23%;width:92.76%;height:100.0%'><span 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onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s45058" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s45059" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:102pt;top:126pt;width:546pt;height:356.37= 5pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s45058" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Recall Catalog Data</div> <div v:shape=3D"_x0000_s45059"> <div class=3DB style=3D'position:absolute;top:24.25%;left:15.16%;width:73.= 97%; height:6.0%'>Sub-samples: </div> <div class=3DB1 style=3D'position:absolute;top:31.5%;left:20.22%;width:68.= 91%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.34%;width:95.= 65%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.54%'>&= #8211;</span>Equally sized sub-samples </span></div> <div class=3DB1 style=3D'position:absolute;top:38.0%;left:20.22%;width:71.= 16%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.21%;width:95.= 52%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.4%'>&#= 8211;</span>“True” model: all 1.7 million customers </span></div> <div class=3DB1 style=3D'position:absolute;top:44.25%;left:20.22%;width:68= .91%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.34%;width:95.= 65%; height:100.0%'><span style=3D'visibility:hidden'><span class=3DB1B style=3D'position:absolute;left:-4.54%'>–</span></span> </span><= /div> <div class=3DB style=3D'position:absolute;top:51.0%;left:15.16%;width:73.9= 7%; height:6.0%'>Confidence intervals: </div> <div class=3DB1 style=3D'position:absolute;top:58.25%;left:20.22%;width:73= .22%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.09%;width:95.= 65%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.27%'>&= #8211;</span>Confidence bound for each sub-sample </span></div> <div class=3DB1 style=3D'position:absolute;top:64.5%;left:20.22%;width:71.= 34%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.19%;width:95.= 8%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.38%'>&= #8211;</span>% times <i>True</i> value falls within bounds </span></div> <div class=3DB1 style=3D'position:absolute;top:71.0%;left:20.22%;width:72.= 28%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.14%;width:95.= 59%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.33%'>&= #8211;</span>Repeat 10 times (redraw sub-samples)</span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" 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<![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0048.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Control: Improving the Polic= y"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"58"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 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v:shape=3D"_x0000_s59394" class=3DT style=3D'position:absolute;top:13= .0%; left:5.99%;width:88.2%;height:6.75%'>Control: Improving the Policy</div> <div v:shape=3D"_x0000_s59395"> <div class=3DB style=3D'position:absolute;top:31.75%;left:10.11%;width:88.= 57%; height:6.0%'>Compare current policy vs. “optimal” policy <= /div> <div class=3DB style=3D'position:absolute;top:39.25%;left:10.11%;width:80.= 71%; height:6.0%'> </div> <div class=3DB style=3D'position:absolute;top:46.75%;left:10.11%;width:95.= 88%; height:6.0%'>Optimal policy contains another source of bias </div> <div class=3DB1 style=3D'position:absolute;top:53.75%;left:15.16%;width:75= .65%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.2%;width:95.7= 9%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.39%'>&= #8211;</span>Optimization favors positive errors </span></div> <div class=3DB1><span style=3D'position:absolute;top:60.25%;left:18.35%; width:82.58%;height:5.25%'><span class=3DB1B style=3D'position:absolute; left:-3.85%'>–</span>Errors in rewards and transitions are no longer= </span><span style=3D'position:absolute;top:65.5%;left:18.35%;width:72.47%;height:5.25%= '>unbiased </span></div> <div class=3DB1><span style=3D'position:absolute;top:72.0%;left:18.35%;wid= th:82.58%; height:5.25%'><span class=3DB1B style=3D'position:absolute;left:-3.85%'>&#= 8211;</span>Bias in rewards and transitions leads to bias in </span><span style=3D'position= :absolute; top:77.25%;left:18.35%;width:72.47%;height:5.25%'>value function estimates </span></div> <div class=3DB style=3D'position:absolute;top:84.0%;left:10.11%;width:80.7= 1%; height:6.0%'><span style=3D'mso-special-format:lastCR;display:none'> <= /span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0038.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0038.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Take home message"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"48"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s49154" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s49155" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:132pt;top:126pt;width:510pt;height:356.37= 5pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s49154" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Take home message</div> <div v:shape=3D"_x0000_s49155"> <div class=3DB><span style=3D'position:absolute;top:24.25%;left:19.28%;wid= th:82.02%; height:5.25%'><span style=3D'font-size:86%'><b>Small error in parameter es= timate may lead </b></span></span><span style=3D'position:absolute;top:29.5%; left:23.03%;width:65.16%;height:5.25%'><span style=3D'font-size:86%'><b>to= big error in value function </b></span></span></div> <div class=3DB style=3D'position:absolute;top:36.0%;left:19.28%;width:68.9= 1%; height:5.25%'><span style=3D'font-size:86%;display:none'><b> </b></spa= n></div> <div class=3DB style=3D'position:absolute;top:42.5%;left:19.28%;width:68.9= 1%; height:5.25%'><span style=3D'font-size:86%'><b>Illustrate magnitudes <= /b></span></div> <div class=3DB1 style=3D'position:absolute;top:48.5%;left:24.34%;width:64.= 04%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.97%;width:95.0= 2%; height:100.0%'><span style=3D'font-size:83%'><span class=3DB1B style=3D'po= sition: absolute;left:-5.23%'>–</span></span><span style=3D'font-size:83%'>V= ariance is large </span></span></div> <div class=3DB1 style=3D'position:absolute;top:54.0%;left:24.34%;width:73.= 97%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.3%;width:95.94= %; height:100.0%'><span style=3D'font-size:83%'><span class=3DB1B style=3D'po= sition: absolute;left:-4.48%'>–</span></span><span style=3D'font-size:83%'>B= ias is smaller but larger for the control problem </span></span></div> <div class=3DB1 style=3D'position:absolute;top:59.25%;left:24.34%;width:64= .04%; height:4.5%'><span style=3D'mso-special-format:nobullet;display:none'>R= 11;</span><span style=3D'font-size:83%;display:none'><b> </b></span></div> <div class=3DB style=3D'position:absolute;top:64.75%;left:19.28%;width:68.= 91%; height:5.25%'><span style=3D'font-size:86%'><b>Closed form expressions</b>= </span><span style=3D'font-size:86%'> </span></div> <div class=3DB1 style=3D'position:absolute;top:71.0%;left:24.34%;width:64.= 04%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.97%;width:95.0= 2%; height:100.0%'><span style=3D'font-size:83%'><span class=3DB1B style=3D'po= sition: absolute;left:-5.23%'>–</span></span><span style=3D'font-size:83%'>B= ias and variance </span></span></div> <div class=3DB1 style=3D'position:absolute;top:76.25%;left:24.34%;width:64= .04%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.97%;width:95.0= 2%; height:100.0%'><span style=3D'font-size:83%'><span class=3DB1B style=3D'po= sition: absolute;left:-5.23%'>–</span></span><span style=3D'font-size:83%'>C= onfidence bounds </span></span></div> <div class=3DB1 style=3D'position:absolute;top:81.5%;left:24.34%;width:64.= 04%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.97%;width:95.0= 2%; height:100.0%'><span style=3D'font-size:83%'><span class=3DB1B style=3D'po= sition: absolute;left:-5.23%'>–</span></span><span style=3D'font-size:83%'>C= ompare current and “optimal” policies </span></span></div> <div class=3DB1 style=3D'position:absolute;top:87.0%;left:24.34%;width:64.= 04%; height:4.5%'><span style=3D'position:absolute;top:0%;left:4.97%;width:95.0= 2%; height:100.0%'><span style=3D'font-size:83%;visibility:hidden'><span class= =3DB1B style=3D'position:absolute;left:-5.23%'>–</span></span><span style=3D'font-size:83%;display:none'> </span></span></div> <div class=3DB style=3D'position:absolute;top:92.5%;left:19.28%;width:68.9= 1%; height:5.25%'><span style=3D'font-size:86%;mso-special-format:lastCR;displ= ay: none'><b> </b></span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0039.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0039.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: Future Research"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"49"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s50178" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s50179" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:1in;top:126pt;width:612pt;height:356.375p= t'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s50178" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>Future Research</div> <div v:shape=3D"_x0000_s50179"> <div class=3DB style=3D'position:absolute;top:24.25%;left:11.04%;width:90.= 82%; height:6.0%'>Use variance estimates to guide exploration </div> <div class=3DB style=3D'position:absolute;top:31.75%;left:11.04%;width:83.= 14%; height:6.0%'> </div> <div class=3DB style=3D'position:absolute;top:39.25%;left:11.04%;width:83.= 14%; height:6.0%'>Improve control algorithms </div> <div class=3DB style=3D'position:absolute;top:46.75%;left:11.04%;width:83.= 14%; height:6.0%'> </div> <div class=3DB style=3D'position:absolute;top:54.25%;left:11.04%;width:95.= 13%; height:6.0%'>Transitions derived from model free estimates </div> <div class=3DB1><span style=3D'position:absolute;top:61.5%;left:19.1%;widt= h:88.38%; height:5.25%'><span class=3DB1B style=3D'position:absolute;left:-3.6%'>= 211;</span>Parametric approaches (model belongs to a small </span><span style=3D'position:absolu= te; top:66.75%;left:19.1%;width:75.09%;height:5.25%'>family) </span></div> <div class=3DB1 style=3D'position:absolute;top:73.25%;left:15.91%;width:78= .08%; height:5.25%'><span style=3D'position:absolute;top:0%;left:4.07%;width:96.= 16%; height:100.0%'><span class=3DB1B style=3D'position:absolute;left:-4.23%'>&= #8211;</span>Nested estimation</span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0050.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0050.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004: The Control Problem"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"61"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><p:shaperange href=3D"master03.xml#_x0000_m1026"/><v:shape id=3D"_x0000_s62466" type=3D= "#_x0000_m1026" style=3D'position:absolute;left:36pt;top:42pt;width:9in;height:90pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"title"/></v:shape><p:shaperange href=3D"master03.xml#_x0000_m1027"/><v:shape id=3D"_x0000_s62467" type=3D= "#_x0000_m1027" style=3D'position:absolute;left:66pt;top:171.625pt;width:594pt;height:356= .375pt'> <v:fill o:detectmouseclick=3D"f"/> <v:stroke o:forcedash=3D"f"/> <o:lock v:ext=3D"edit" text=3D"f"/> <p:placeholder type=3D"body" position=3D"1"/></v:shape> <div v:shape=3D"_x0000_s62466" class=3DT style=3D'position:absolute;top:13= .0%; left:5.99%;width:88.2%;height:6.75%'>The Control Problem</div> <div v:shape=3D"_x0000_s62467"> <div class=3DB2 style=3D'mso-margin-left-alt:864'></div> <div class=3DB3 style=3D'mso-margin-left-alt:1152'></div> <div class=3DB4 style=3D'mso-margin-left-alt:1440'></div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:32.7= 5%; left:10.11%;width:80.71%;height:6.0%'>Cross-validate to remove bias </= div> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:40.= 0%; left:15.16%;width:75.65%;height:5.25%'><span style=3D'position:absolute; top:0%;left:6.68%;width:93.31%;height:100.0%'><span style=3D'visibility:hi= dden'><span style=3D'mso-special-format:bullet;position:absolute;left:-7.16%'>–<= /span></span> </span></div> <div class=3DB style=3D'mso-margin-left-alt:336'><span style=3D'position:a= bsolute; top:46.75%;left:15.91%;width:82.39%;height:6.0%'><span style=3D'mso-specia= l-format: "numbullet3\,1";position:absolute;left:-7.04%'>1.</span>Derive optimal pol= icy using a calibration </span><span style=3D'position:absolute;top:53.0%; left:15.91%;width:74.9%;height:6.0%'>sample </span></div> <div class=3DB1 style=3D'mso-margin-left-alt:576;position:absolute;top:60.= 0%; left:15.16%;width:75.65%;height:5.25%'><span style=3D'position:absolute; top:0%;left:6.68%;width:93.31%;height:100.0%'><span style=3D'visibility:hi= dden'><span style=3D'mso-special-format:bullet;position:absolute;left:-7.16%'>–<= /span></span> </span></div> <div class=3DB style=3D'mso-margin-left-alt:336;position:absolute;top:66.7= 5%; left:10.11%;width:87.45%;height:6.0%'><span style=3D'position:absolute; top:0%;left:6.63%;width:93.36%;height:100.0%'><span style=3D'mso-special-f= ormat: "numbullet3\,1";position:absolute;left:-7.11%'>2.</span>Evaluate policy us= ing validation sample</span></div> </div> </p:slide></div> </body> </html> ------=_NextPart_01C48638.3C956210 Content-Location: file:///C:/8EB2BDB1/BiasVarianceICML1_files/slide0045.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" <html xmlns:v=3D"urn:schemas-microsoft-com:vml" xmlns:o=3D"urn:schemas-microsoft-com:office:office" xmlns:p=3D"urn:schemas-microsoft-com:office:powerpoint" xmlns:oa=3D"urn:schemas-microsoft-com:office:activation" xmlns=3D"http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=3DContent-Type content=3D"text/html; charset=3Dus-ascii"> <meta name=3DProgId content=3DPowerPoint.Slide> <meta name=3DGenerator content=3D"Microsoft PowerPoint 10"> <link id=3DMain-File rel=3DMain-File href=3D"../BiasVarianceICML1.htm"> <link rel=3DPreview href=3Dpreview.wmf> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} p\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} v\:textbox {display:none;} </style> <![endif]--> <title>Bias and Variance in Value Function Estimates - ICML/Shie Mannor <= /title> <![if IE]> <base href=3D"file:///C:\8EB2BDB1\BiasVarianceICML1_files\slide0045.htm" id=3D"webarch_temp_base_tag"> <![endif]> <meta name=3DDescription content=3D"8/19/2004"> <link rel=3DStylesheet href=3D"master03_stylesheet.css"> <![if !ppt]> <style media=3Dprint> <!--.sld {left:0px !important; width:6.0in !important; height:4.5in !important; font-size:107% !important;} --> </style> <script src=3Dscript.js></script><script><!-- if( !IsNts() ) Redirect( "PPTSld" ); //--></script><!--[if vml]><script>g_vml =3D 1; </script><![endif]--><![endif]><o:shapelayout v:ext=3D"edit"> <o:idmap v:ext=3D"edit" data=3D"55"/> </o:shapelayout> </head> <body lang=3DEN-US style=3D'margin:0px;background-color:black' onclick=3D"DocumentOnClick()" onresize=3D"_RSW()" onload=3D"LoadSld()" onkeypress=3D"_KPH()"> <div id=3DSlideObj class=3Dsld style=3D'position:absolute;top:0px;left:0px; width:534px;height:400px;font-size:16px;background-color:white;clip:rect(0%= , 101%, 101%, 0%); visibility:hidden'><p:slide coordsize=3D"720,540" colors=3D"#FFFFFF,#000000,#808080,#000000,#BBE0E3,#333399,#009999,#99CC00" masterhref=3D"master03.xml"> <p:shaperange href=3D"master03.xml#_x0000_s1025"/><![if !ppt]><p:shaperange href=3D"master03.xml#_x0000_s1028"/><p:shaperange href=3D"master03.xml#_x0000_s1029"/><![endif]><v:rect id=3D"_x0000_s56325" style=3D'position:absolute;left:36pt;top:21.625pt;width:9in;height:90pt; v-text-anchor:middle' filled=3D"f" fillcolor=3D"#bbe0e3 [4]" stroked=3D"f" strokecolor=3D"black [1]"> <v:fill color2=3D"white [0]"/> <v:shadow color=3D"gray [2]"/> </v:rect> <div v:shape=3D"_x0000_s56325" class=3DT style=3D'position:absolute;top:9.= 25%; left:5.99%;width:88.2%;height:6.75%'>The Control Problem:<span style=3D'mso-spacerun:yes'>  </span>Bias</div> <v:shapetype id=3D"_x0000_t75" coordsize=3D"21600,21600" o:spt=3D"75" o:preferrelative=3D"t" path=3D"m@4@5l@4@11@9@11@9@5xe" filled=3D"f" strok= ed=3D"f"> <v:stroke joinstyle=3D"miter"/> <v:formulas> <v:f eqn=3D"if lineDrawn pixelLineWidth 0"/> <v:f eqn=3D"sum @0 1 0"/> <v:f eqn=3D"sum 0 0 @1"/> <v:f eqn=3D"prod @2 1 2"/> <v:f eqn=3D"prod @3 21600 pixelWidth"/> <v:f eqn=3D"prod @3 21600 pixelHeight"/> <v:f eqn=3D"sum @0 0 1"/> <v:f eqn=3D"prod @6 1 2"/> <v:f eqn=3D"prod @7 21600 pixelWidth"/> <v:f eqn=3D"sum @8 21600 0"/> <v:f eqn=3D"prod @7 21600 pixelHeight"/> <v:f eqn=3D"sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok=3D"f" gradientshapeok=3D"t" o:connecttype=3D"rect"/> <o:lock v:ext=3D"edit" aspectratio=3D"t"/> </v:shapetype><v:shape id=3D"_x0000_s56326" type=3D"#_x0000_t75" 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function LoadSld() { var sld=3DGetObj("SlideObj") if( !g_supportsPPTHTML ) { sld.style.visibility=3D"visible" return } if( MakeNotesVis() ) return runAnimations =3D _InitAnimations(); if( IsWin("PPTSld") ) parent.SldUpdated(GetSldId()) g_origSz=3DparseInt(SlideObj.style.fontSize) g_origH=3Dsld.style.posHeight g_origW=3Dsld.style.posWidth g_scaleHyperlinks=3D(document.all.tags("AREA").length>0) if( g_scaleHyperlinks ) InitHLinkArray() if( g_scaleInFrame||(IsWin("PPTSld") && parent.IsFullScrMode() ) ) document.body.scroll=3D"no" _RSW() if( IsWin("PPTSld") && parent.IsFullScrMode() ) FullScrInit(); =09 MakeSldVis(); ChkAutoAdv() if( runAnimations ) { if( document.all("NSPlay") ) document.all("NSPlay").autoStart =3D false; if( sld.filters && sld.filters.revealtrans ) setTimeout( "document.body.start()", sld.filters.revealtrans.duration * = 1000 ); else document.body.start(); } } function MakeSldVis() { var fTrans=3Dg_showAnimation && SldHasTrans() if( fTrans ) { if( g_bgSound ) { 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} function SetHasNts( fVal ) { if( gHasNts !=3D fVal ) { gHasNts=3DfVal UpdNavPane() } } function ToggleOtlText() { gOtlTxtExp=3D!gOtlTxtExp UpdOtlPane() } function ToggleOtlPane() { frmset=3Ddocument.all("PPTHorizAdjust") frm=3Ddocument.all("PPTOtl") if( gOtlOpen ) frmset.cols=3D"*,100%" else frmset.cols=3D"25%,*" gOtlOpen=3D!gOtlOpen frm.noResize=3D!frm.noResize UpdOtNavPane() } function ToggleNtsPane() { frmset=3Ddocument.all("PPTVertAdjust") frm=3Ddocument.all("PPTNts") if( gNtsOpen ) frmset.rows=3D"100%,*" else frmset.rows=3D"*,20%" gNtsOpen=3D!gNtsOpen UpdNtsPane() } function ClearMedia() { if (PPTSld.pptSound) PPTSld.pptSound.loop =3D 0; } function FullScreen() { if ( PPTSld.g_animUseRuntime ) PPTSld.document.body.pause(); ClearMedia(); var href =3D ( document.location.protocol =3D=3D 'mhtml:') ? FULLSCR_HREF = : FULLSCR_HREF+"#"+GetHrefObj(gCurSld).mSldHref; if(PPTNav.event.ctrlKey) { var w =3D (window.screen.availWidth * 1.0) / 2.0 var h =3D w * (PPTSld.g_origH * 1.0) / PPTSld.g_origW win =3D window.open( MHTMLPrefix+href,null,"toolbar=3D0,resizable=3D1,top= =3D0,left=3D0," + "width=3D"+ w + ",height=3D" + h ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } else { win =3D window.open( MHTMLPrefix+href,null,"fullscreen=3Dyes" ); if( PPTSld.g_animUseRuntime ) win.document.body.PPTSldFrameset=3Dwindow; } } function ToggleVNarration() { rObj=3DPPTSld.document.all("NSPlay") if( rObj && !PPTSld.g_animUseRuntime ) { if( (rObj.playState =3D=3D 1)||(rObj.playState =3D=3D 0) ) rObj.Play() else if( rObj.playState =3D=3D 2 ) rObj.Pause() else return; } else if( PPTSld.g_animUseRuntime ) { narObj =3D PPTSld.document.all("narrationID") if( narObj ) narObj.togglePause() } } function GetCurSldNum() { obj=3DGetHrefObj(gCurSld) if( obj.mOrigVis =3D=3D 1 ) return obj.mSldIdx else return gCurSld } function GetNumSlds() { if( GetHrefObj(gCurSld).mOrigVis =3D=3D 1 ) return GetSldList().mNumVisSlds; else return GetSldList().mList.length } function GetSldNum( href ) { for(ii=3D0; ii<GetSldList().mList.length; ii++) { if ( GetSldList().mList[ii].mSldHref =3D=3D href ) return ii+1 } return 1 } function GetHrefObj( sldIdx ){ return GetSldList().mList[sldIdx-1] } function IsFramesMode(){ return ( SCREEN_MODE =3D=3D "Frames" ) } function IsFullScrMode(){ return ( SCREEN_MODE =3D=3D "FullScreen" ) } function GoToNextSld() { ii=3DgCurSld + 1 if( GetHrefObj( ii-1 ).mOrigVis =3D=3D 0 ) { if( ii<=3DGetSldList().mList.length ) { obj=3DGetHrefObj(ii) obj.mVis=3D1 GoToSldNum(ii) return } } else { obj=3DGetHrefObj( ii ) while ( obj && ( obj.mOrigVis =3D=3D 0 ) ) obj=3DGetHrefObj(++ii) if( obj && obj.mOrigVis ) { GoToSldNum(ii) return } } if( gSldStack.length > 1 ) PopSldList(); else if( !IsFramesMode() ) { if( gLoopCont ) GoToFirst() else EndShow() } } function GoToPrevSld() { ii=3DgCurSld-1 if( ii > 0 ) { obj=3DGetHrefObj(ii) while ( obj && ( obj.mVis =3D=3D 0 ) && ( ii>0 ) ) obj=3DGetHrefObj(--ii) if( ii =3D=3D 0 ) ii=3D1 GoToSldNum(ii) } } function GoToFirst(){ GoToSld( GetHrefObj(1).mSldHref ) } function GoToLast() { ii=3DGetSldList().mList.length if( ii !=3D gCurSld ) GoToSld( GetHrefObj(ii).mSldHref ) } function GoToSldNum( num ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue obj =3D GetHrefObj( num ) obj.mVis=3D1 gPrevSld=3DgCurSld gCurSld =3D num; PPTSld.location.replace(MHTMLPrefix+obj.mSldHref) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function GoToSld( href ) { if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue GetHrefObj( GetSldNum(href) ).mVis=3D1 PPTSld.location.replace(MHTMLPrefix+href) } function SldUpdated( id ) { if( id =3D=3D GetHrefObj(gCurSld).mSldHref ) return gPrevSld=3DgCurSld gCurSld=3DGetSldNum(id) if( IsFramesMode() ) { UpdNavPane(); UpdOtlPane(); UpdNtsPane() } } function PrevSldViewed(){ GoToSld( GetHrefObj(gPrevSld).mSldHref ) } function HasPrevSld() { return ( gIsEndShow || ( gCurSld !=3D 1 && GetHrefO= bj( gCurSld-1 ).mVis =3D=3D 1 )||( GetCurSldNum() > 1 ) ) } function HasNextSld() { return (GetCurSldNum() !=3D GetNumSlds()) } function CloseWindow() { if( HideMenu() ) return; var event =3D PPTSld.event; if( !IsFramesMode() && event && (event.keyCode=3D=3D27 || event.keyCode=3D= =3D32 || event.type=3D=3D"click" ) ) window.close( self ); CatchNumKeys( self, event ); } function Unload() { gIsEndShow=3D0; } function SetupEndShow() { gIsEndShow=3D1; PPTSld.document.body.scroll=3D"no"; PPTSld.document.onkeypress=3DCloseWindow; PPTSld.document.onclick=3DCloseWindow; PPTSld.document.oncontextmenu=3D_CM; } function EndShow() { if( IsFramesMode() ) return if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue doc=3DPPTSld.document doc.open() doc.writeln('<html><body bgcolor=3Dblack onload=3Dparent.SetupEndShow() on= unload=3Dparent.Unload()><center><p><font face=3DTahoma color=3Dwhite size= =3D2><br><b>' + ENDSHOW_MESG + '</b></font></p></center></body></html>') doc.close() } function SetSldVisited(){ GetSldList().mList[gCurSld-1].mVisited=3Dtrue } function IsSldVisited(){ return GetSldList().mList[gCurSld-1].mVisited } function hrefList( sldHref, visible, advDelay, advClk ) { this.mSldHref=3D this.mNtsHref =3D sldHref this.mOrigVis=3D this.mVis =3D visible this.mVisited=3D false this.mAdvDelay=3D advDelay this.mAdvOnClk=3D advClk } function SldList(arr,curSld,fEnd) { this.mCurSld =3D curSld; this.mList =3D new Array(); var idx =3D 1; for(ii=3D0;ii<arr.length;ii++) { this.mList[ii] =3D new hrefList( arr[ii].mSldHref, arr[ii].mOrigVis, arr[= ii].mAdvDelay, arr[ii].mAdvOnClk ); if( arr[ii].mOrigVis ) this.mList[ii].mSldIdx =3D idx++; } this.mNumVisSlds =3D idx-1; this.fEndShow =3D fEnd; } function GetSldList() { return gSldStack[gSldStack.length-1] } function GetCurSld() { return parent.GetSldList().mList[parent.gCurSld - 1]= } gSldStack =3D new Array(); gSldStack[0] =3D new SldList(gMainDoc,gCurSld,1) function ViewCustomShow(idx,fEnd) { if( !IsFullScrMode() ) return; var sldList =3D new Array(); var custShow =3D custShowList[idx-1]; var jj =3D 0; for( ii=3D0;ii<custShow.length;ii++ ) { if( custShow[ii] <=3D gMainDoc.length ) sldList[jj++] =3D gMainDoc[custShow[ii]-1]; } if (sldList.length > 0) { PushSldList(sldList,fEnd); gCurSld =3D 1; } else if( PPTSld.event ) PPTSld.event.cancelBubble=3Dtrue } function PushSldList(arr,fEnd) { var ii =3D gSldStack.length; gSldStack[ii] =3D new SldList(arr,gCurSld,fEnd); GoToSld( gSldStack[ii].mList[0].mSldHref ); } function PopSldList() { if (gSldStack[gSldStack.length-1].fEndShow) EndShow() else { gCurSld =3D gSldStack[gSldStack.length-1].mCurSld; gSldStack[gSldStack.length-1] =3D null; gSldStack.length--; var sldList =3D gSldStack[gSldStack.length-1]; GoToSld( sldList.mList[gCurSld - 1].mSldHref ); } } var custShowList=3Dnew Array(); function ImgBtn( oId,bId,w,action ) { var t=3Dthis t.Perform =3D _IBP t.SetActive =3D _IBSetA t.SetInactive=3D _IBSetI t.SetPressed =3D _IBSetP t.SetDisabled=3D _IBSetD t.Enabled =3D _IBSetE t.ChangeIcon =3D null t.UserAction =3D action t.ChgState =3D _IBUI t.mObjId =3D oId t.mBorderId=3D bId t.mWidth =3D w t.mIsOn =3D t.mCurState =3D 0 } function _IBSetA() { if( this.mIsOn ) { obj=3Dthis.ChgState( gHiliteClr,gShadowClr,2 ) obj.style.posTop=3D0 } } function _IBSetI() { if( this.mIsOn ) { obj=3Dthis.ChgState( gFaceClr,gFaceClr,1 ) obj.style.posTop=3D0 } } function _IBSetP() { if( this.mIsOn ) { obj=3Dthis.ChgState( gShadowClr,gHiliteClr,2 ) obj.style.posLeft+=3D1; obj.style.posTop+=3D1 } } function _IBSetD() { obj=3Dthis.ChgState( gFaceClr,gFaceClr,0 ) obj.style.posTop=3D0 } function _IBSetE( state ) { var t=3Dthis GetObj( t.mBorderId ).style.visibility=3D"visible" if( state !=3D t.mIsOn ) { t.mIsOn=3Dstate if( state ) t.SetInactive() else t.SetDisabled() } } function _IBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( t.ChangeIcon ) { obj=3DGetObj(t.mObjId) if( t.ChangeIcon() ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-4)*t.mWidth else obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-0)*t.mWidth } t.SetActive() } } function _IBUI( clr1,clr2,nextState ) { var t=3Dthis SetBorder( GetObj( t.mBorderId ),clr1,clr2 ) obj=3DGetObj( t.mObjId ) obj.style.posLeft=3Dobj.style.posLeft+(t.mCurState-nextState)*t.mWidth-obj= .style.posTop t.mCurState=3DnextState return obj } function TxtBtn( oId,oeId,action,chkState ) { var t=3Dthis t.Perform =3D _TBP t.SetActive =3D _TBSetA t.SetInactive=3D _TBSetI t.SetPressed =3D _TBSetP t.SetDisabled=3D _TBSetD t.SetEnabled =3D _TBSetE t.GetState =3D chkState t.UserAction =3D action t.ChgState =3D _TBUI t.mObjId =3D oId t.m_elementsId=3D oeId t.mIsOn =3D 1 } function _TBSetA() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gHiliteClr,gShadowClr,0,0 ) } function _TBSetI() { var t=3Dthis if( t.mIsOn && !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) } function _TBSetP() { if( this.mIsOn ) this.ChgState( gShadowClr,gHiliteClr,1,1 ) } function _TBSetD() { this.ChgState( gFaceClr,gFaceClr,0,0 ) this.mIsOn =3D 0 } function _TBSetE() { var t=3Dthis if( !t.GetState() ) t.ChgState( gFaceClr,gFaceClr,0,0 ) else t.ChgState( gShadowClr,gHiliteClr,1,1 ) t.mIsOn =3D 1 } function _TBP() { var t=3Dthis if( t.mIsOn ) { if( t.UserAction !=3D null ) t.UserAction() if( !t.GetState ) return if( t.GetState() ) t.SetPressed() else t.SetActive() } } function _TBUI( clr1,clr2,lOffset,tOffset ) { SetBorder( GetObj( this.mObjId ),clr1,clr2 ) Offset( GetObj( this.m_elementsId ),lOffset,tOffset ) } function Offset( obj, top, left ){ obj.style.top=3Dtop; obj.style.left=3Dle= ft } function SetBorder( obj, upperLeft, lowerRight ) { s=3Dobj.style; s.borderStyle =3D "solid" s.borderWidth =3D 1 s.borderLeftColor =3D s.borderTopColor =3D upperLeft s.borderBottomColor=3D s.borderRightColor =3D lowerRight } function GetBtnObj(){ return gBtnArr[window.event.srcElement.id] } function BtnOnOver(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetActive() } function BtnOnDown(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetPressed() } function BtnOnOut(){ b=3DGetBtnObj(); if( b !=3D null ) b.SetInactive() } function BtnOnUp() { b=3DGetBtnObj() if( b !=3D null ) b.Perform() else Upd() } function GetNtsState(){ return parent.gNtsOpen } function GetOtlState(){ return parent.gOtlOpen } function GetOtlTxtState(){ return parent.gOtlTxtExp } function NtsBtnSetFlag( fVal ) { s=3Ddocument.all.item( this.m_flagId ).style s.display=3D"none" if( fVal ) s.display=3D"" else s.display=3D"none" } function _BSetA_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etActive() } function _BSetI_Border(){ b =3D gBtnArr[this.mObjId]; if( b !=3D null ) b.S= etInactive() } var gHiliteClr=3D"THREEDHIGHLIGHT",gShadowClr=3D"THREEDSHADOW",gFaceClr=3D"= THREEDFACE" var gBtnArr =3D new Array() gBtnArr["nb_otl"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOtlPan= e,GetOtlState ) gBtnArr["nb_otlElem"] =3D new TxtBtn( "nb_otl","nb_otlElem",parent.ToggleOt= lPane,GetOtlState ) gBtnArr["nb_nts"] =3D new TxtBtn( "nb_nts","nb_ntsElem",parent.ToggleNtsPan= e,GetNtsState ) gBtnArr["nb_prev"]=3D new ImgBtn( "nb_prev","nb_prevBorder",30,parent.GoToP= revSld ) gBtnArr["nb_next"]=3D new ImgBtn( "nb_next","nb_nextBorder",30,parent.GoToN= extSld ) gBtnArr["nb_sldshw"]=3D new ImgBtn( "nb_sldshw","nb_sldshwBorder",18,parent= .FullScreen ) gBtnArr["nb_sldshwBorder"] =3D new TxtBtn( "nb_sldshw","nb_sldshwBorder",pa= rent.FullScreen,null ) gBtnArr["nb_sldshwBorder"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwBorder"].SetInactive =3D _BSetI_Border; gBtnArr["nb_sldshwText"] =3D new TxtBtn( "nb_sldshw","nb_sldshwText",parent= .FullScreen,null ) gBtnArr["nb_sldshwText"].SetActive =3D _BSetA_Border; gBtnArr["nb_sldshwText"].SetInactive =3D _BSetI_Border; gBtnArr["nb_voice"] =3D new ImgBtn( "nb_voice","nb_voiceBorder",18,parent.= ToggleVNarration ) gBtnArr["nb_otlTxt"]=3D new ImgBtn( "nb_otlTxt","nb_otlTxtBorder",23,parent= .ToggleOtlText ) gBtnArr["nb_nts"].m_flagId=3D "notes_flag" gBtnArr["nb_nts"].SetFlag =3D NtsBtnSetFlag gBtnArr["nb_otlTxt"].ChangeIcon=3D GetOtlTxtState var sNext=3D"Next",sPrev=3D"Previous",sEnd=3D"End Show",sFont=3D"Arial",sAr= row=3D"Arrow",sFreeform=3D"Freeform",sRect=3D"Rectangle",sOval=3D"Oval" function ShowMenu() { BuildMenu(); var doc=3DPPTSld.document.body,x=3DPPTSld.event.clientX+doc.scrollLeft,y= =3DPPTSld.event.clientY+doc.scrollTop m =3D PPTSld.document.all.item("ctxtmenu") m.style.pixelLeft=3Dx if( (x+m.scrollWidth > doc.clientWidth)&&(x-m.scrollWidth > 0) ) m.style.pixelLeft=3Dx-m.scrollWidth m.style.pixelTop=3Dy if( (y+m.scrollHeight > doc.clientHeight)&&(y-m.scrollHeight > 0) ) m.style.pixelTop=3Dy-m.scrollHeight m.style.display=3D"" } function _CM() { if( !parent.IsFullScrMode() ) return; if(!PPTSld.event.ctrlKey) { ShowMenu() return false } else HideMenu() } function BuildMenu() { if( PPTSld.document.all.item("ctxtmenu") ) return var mObj=3DCreateItem( PPTSld.document.body ) mObj.id=3D"ctxtmenu" mObj.style.visibility=3D"hidden" var s=3DmObj.style s.position=3D"absolute" s.cursor=3D"default" s.width=3D"120px" SetCMBorder(mObj,"menu","black") var iObj=3DCreateItem( mObj ) SetCMBorder( iObj, "threedhighlight","threedshadow" ) iObj.style.padding=3D2 CreateMenuItem( iObj,sNext,M_GoNextSld,M_True ) CreateMenuItem( iObj,sPrev,M_GoPrevSld,M_HasPrevSld ) CreateSeparator( iObj ) CreateMenuItem( iObj,sEnd,M_End,M_True ) mObj.style.visibility=3D"visible" } function Cancel() { window.event.cancelBubble=3Dtrue; window.event.returnVa= lue=3Dfalse } function Highlight() { ChangeClr("activecaption","threedhighlight") } function Deselect() { ChangeClr("threedface","menutext") } function Perform() { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) e.Action() else PPTSld.event.cancelBubble=3Dtrue } function ChangeClr( bg,clr ) { e=3DPPTSld.event.srcElement if( e.type=3D=3D"menuitem" && e.IsActive() ) { e.style.backgroundColor=3Dbg e.style.color=3Dclr } } function M_HasPrevSld() { return( parent.HasPrevSld() ) } function M_GoNextSld() { if( gIsEndShow ) M_End(); 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Tsitsiklis (MIT)</a></font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>2</font></div> </td> <td> <div id=3DPPTP2 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0051.htm');" id=3DPPTL2>Estimating Value Function from Data</a></font></div> <div id=3DPPTC2 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>So you have collected your data</li> <li>Estimate rewards and transitions</li> <ul> <li>For each action in each state</li> <br> <br> </ul> <li>Estimate value function for a fixed policy</li> <br> <br> <li>Improve policy</li> <br> <br> <li>How good is your estimate? (confidence)</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>3</font></div> </td> <td> <div id=3DPPTP3 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0052.htm');" id=3DPPTL3>Estimating Value Functions</a></font></div> <div id=3DPPTC3 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Value function:</li> <br> <br> <br> <br> <li>Rearranging, we obtain:</li> <br> <br> <br> <br> <br> <li>Errors in R and P lead to errors in V.</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>4</font></div> </td> <td> <div id=3DPPTP4 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0053.htm');" id=3DPPTL4>Parametric vs. I= nternal Variance</a></font></div> <div id=3DPPTC4 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Internal:</li> <br> <br> <br> <li>Parametric:</li> <br> <br> <br> <li>Different entities, used for different things, different probabilist= ic setup</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>5</font></div> </td> <td> <div id=3DPPTP5 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0025.htm');" id=3DPPTL5>Illustration<br> The Catalog Circulation Problem</a></font></div> <div id=3DPPTC5 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Women’s clothing retailer</li> <li>1.7 million customers</li> <li>Entire purchase history</li> <li>6 years of mailing history</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>6</font></div> </td> <td> <div id=3DPPTP6 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0015.htm');" id=3DPPTL6>Illustration</a>= </font></div> <div id=3DPPTC6 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>64 states:</li> <ul> <li>Recency, Frequency and Monetary Value</li> <li>Quartile (4<sup>3</sup>)</li> <br> <br> </ul> <li>250 Sub-samples:</li> <ul> <li>6,880 and 657,000 observations in each</li> <li>10,000 / reward and 1,400 / transition</li> </ul> <br> <li>“True” model</li> <ul> <li>All 1.7 million customers</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>7</font></div> </td> <td> <div id=3DPPTP7 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0018.htm');" id=3DPPTL7>Estimating Value Functions</a></font></div> <div id=3DPPTC7 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>64 states (RFM)</li> <ul> <br> <br> </ul> <li>Estimate rewards and transitions</li> <ul> <li>250 parameter estimates</li> <br> <br> </ul> <li>Policy evaluation</li> <ul> <li>250 value function estimates</li> <li>Compare with “True” value function</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>8</font></div> </td> <td> <div id=3DPPTP8 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0026.htm');" id=3DPPTL8><span style=3D'mso-spacerun:yes'> </span></a></font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>9</font></div> </td> <td> <div id=3DPPTP9 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0021.htm');" id=3DPPTL9>Bias and Varianc= e</a></font></div> <div id=3DPPTC9 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Bias:</li> <ul> <li>True mean:<span style=3D'mso-spacerun:yes'>  </span>$79.01</li> <li>Mean across sub-samples:<span style=3D'mso-spacerun:yes'>  </span>$79.20</li> <li>Standard deviation:<span style=3D'mso-spacerun:yes'>  </span>$= 4.95</li> <br> <br> </ul> <li>Variance:</li> <ul> <li>95% confidence interval:<span style=3D'mso-spacerun:yes'>  </span>$69.11 to $88.91</li> <br> <br> </ul> <li>Robustness:</li> <ul> <li>Vary size of sub-samples</li> <li>Vary discount rate</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>10</font></div> </td> <td> <div id=3DPPTP10 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0027.htm');" id=3DPPTL10><span style=3D'mso-spacerun:yes'> </span></a></font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>11</font></div> </td> <td> <div id=3DPPTP11 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0022.htm');" id=3DPPTL11>Analysis: Fixed= Policy</a></font></div> <div id=3DPPTC11 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Derive 1st and 2nd moments of V</li> <br> <br> <br> <li>Write:</li> <br> <br> <br> <li>Assume:</li> <ul> <li>Use sample values of the model parameters</li> <li>2nd order approximation</li> <br> <br> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>12</font></div> </td> <td> <div id=3DPPTP12 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0054.htm');" id=3DPPTL12>Analysis</a></f= ont></div> <div id=3DPPTC12 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Two approaches:</li> <li>Classical – Derive expressions using true P and R and then substitute them.</li> <br> <br> <li>Bayesian – Assume comfortable priors (Dirichlet and Gaussian)<= /li> <br> <br> <li>Same expressions unless there is a meaningful prior</li> <br> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>13</font></div> </td> <td> <div id=3DPPTP13 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0028.htm');" id=3DPPTL13>Validation</a><= /font></div> <div id=3DPPTC13 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Check accuracy of confidence intervals</li> <br> <br> <li>Evaluate assumptions</li> <ul> <li>Use of sample model parameters</li> <li>2<sup>nd</sup> order approximation</li> <br> <br> </ul> <li>Control problem</li> <ul> <li>Policy no longer fixed</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>14</font></div> </td> <td> <div id=3DPPTP14 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0035.htm');" id=3DPPTL14>Recall Catalog = Data</a></font></div> <div id=3DPPTC14 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Sub-samples:</li> <ul> <li>Equally sized sub-samples</li> <li>“True” model: all 1.7 million customers</li> <br> <br> </ul> <li>Confidence intervals:</li> <ul> <li>Confidence bound for each sub-sample</li> <li>% times True value falls within bounds</li> <li>Repeat 10 times (redraw sub-samples)</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>15</font></div> </td> <td> <div id=3DPPTP15 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0036.htm');" id=3DPPTL15><span style=3D'mso-spacerun:yes'> </span></a></font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>16</font></div> </td> <td> <div id=3DPPTP16 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0048.htm');" id=3DPPTL16>Control: Improv= ing the Policy</a></font></div> <div id=3DPPTC16 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Compare current policy vs. “optimal” policy</li> <br> <br> <li>Optimal policy contains another source of bias</li> <ul> <li>Optimization favors positive errors</li> <li>Errors in rewards and transitions are no longer unbiased</li> <li>Bias in rewards and transitions leads to bias in value function estimates</li> </ul> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>17</font></div> </td> <td> <div id=3DPPTP17 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0038.htm');" id=3DPPTL17>Take home messa= ge</a></font></div> <div id=3DPPTC17 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Small error in parameter estimate may lead to big error in value function</li> <br> <br> <li>Illustrate magnitudes</li> <ul> <li>Variance is large</li> <li>Bias is smaller but larger for the control problem</li> <br> <br> </ul> <li>Closed form expressions</li> <ul> <li>Bias and variance</li> <li>Confidence bounds</li> <li>Compare current and “optimal” policies</li> <br> <br> </ul> <br> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>18</font></div> </td> <td> <div id=3DPPTP18 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0039.htm');" id=3DPPTL18>Future Research= </a></font></div> <div id=3DPPTC18 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Use variance estimates to guide exploration</li> <br> <br> <li>Improve control algorithms</li> <br> <br> <li>Transitions derived from model free estimates</li> <ul> <li>Parametric approaches (model belongs to a small family)</li> <li>Nested estimation</li> </ul> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>19</font></div> </td> <td> <div id=3DPPTP19 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0050.htm');" id=3DPPTL19>The Control Pro= blem</a></font></div> <div id=3DPPTC19 class=3DCTxt style=3D'display:none'><font size=3D2> <ul> <li>Cross-validate to remove bias</li> <ul> <br> <br> </ul> <li>Derive optimal policy using a calibration sample</li> <ul> <br> <br> </ul> <li>Evaluate policy using validation sample</li> </ul> </font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>20</font></div> </td> <td> <div id=3DPPTP20 class=3DPTxt><font size=3D2><a href=3D"javascript:GoToSld('slide0045.htm');" id=3DPPTL20><span style=3D'mso-spacerun:yes'> </span></a></font></div> </td> </tr> <tr onmouseover=3D"Over(this)" onmouseout=3D"Out(this)" onclick=3D"Follow(= this)" style=3D'cursor:hand'> <td align=3Dright valign=3Dtop> <div class=3DsldNum><font size=3D2>21</font></div> </td> <td> <div id=3DPPTP21 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