Variable selection for causal inference(Causal inference: What if, Chapter 18)

Keywords: 因果推論, Variable selection（変数選択），Matchine learning（機械学習），Doubly robust estimation（二重ロバスト推定），Over-fitting，Cross-validation，Causal Tree

Transcript

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   ൃදऀ͠ΜΊʔ ᚸ໌ ܓଠ
   
   πΠολʔ!TIJNNFFFF
   σʔλαΠΤϯςΟετ !ੈքۜߦ JOϫγϯτϯ%$
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2. ࠓ೔ͷίϯςϯπ
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3. ࠓ೔ͷίϯςϯπ
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   ҼՌਪ࿦ͱ༧ଌͷҧ͍
   
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   ௐ੔͢Δ͜ͱͰόΠΞεΛൃੜͤ͞Δม਺
   
   $BVTBMJOGFSFODFBOENBDIJOFMFBSOJOH
   ߴ࣍ݩͷڞมྔΛػցֶशͰௐ੔͢Δࡍͷ໰୊఺ͳͲ
   
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   ҼՌਪ࿦ͷม਺બ୒͸΍ͬͺΓ೉͍͠
   
   ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
   

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4. $BVTBMJOGFSFODFWT1SFEJDUJPOަབྷҼࢠ DPOGPVOEFS
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   o &YDIBOHFBCJMJUZΛຬͨͨ͢Ίʹɼσʔλ෼ੳͷதͰશͯͷަབྷҼࢠΛ -
   Λௐ੔͢Δඞཁ͕͋Δɻ
   o ަབྷҼࢠ͕ௐ੔͞ΕͯॳΊͯ BTTPDJBUJPONFBTVSF͕ DBVTBMFGGFDUNFBTVSFͱղऍͰ͖Δɻ
   q 1SFEJDUJPOʹ͓͚ΔަབྷҼࢠ
   o հೖม਺ "
   ͷ֓೦͕ඞཁͳ͍ͷͰɼަབྷҼࢠ -
   Λௐ੔͢Δඞཁੑ͕ͳ͍ɻ
   • ྫ
   ېԎ 9
   ͱମॏݮগ :
   ͷؔ࿈Λ஌Γ͍ͨ
   Ø ަབྷҼࢠΛௐ੔͢Δඞཁ͕ͳ͍ͷͰɼ୯७ʹ9͝ͱͷ:ͷฏۉͷࠩΛݟΕ͹͍͍ɻ
   o ෼ੳϞσϧͰ͸"΍-Λ۠ผͤͣɼಉ͡ڞมྔ 9
   ͱͯ͠ѻ͏ɻ
   E[Y a=1] E[Y a=0] =
   X
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5. $BVTBM NPEFMWTQSFEJDUJWFBTTPDJBUJPOBMNPEFM
   q ͷ෮श
   o 0VUDPNFSFHSFTTJPO͸DBVTBM NPEFM ͱ QSFEJDUJWF NPEFMͰڞʹར༻͞ΕΔ͕ɼ໨త΍ղऍ͕શ͘ҟͳΔɻ
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   o 1SFEJDUJWFNPEFM:ͷ༧ଌ͕໨త
   q 1SFEJDUJWFNPEFMʹ͓͚Δม਺ͷղऍ
   o ྫ
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   o ద౰ͳ༧ଌϞσϧ FHSBOEPNGPSFTU
   ͔ΒɼʮաڈͷೖӃܦݧʯ͕༗༻ͳม਺ͩͱ൑໌
   o 1SFEJDUJWFNPEFM͸ม਺ͷҼՌޮՌΛදݱ͢Δ΋ͷͰ͸ͳ͍ɻ
   Ø ʮকདྷ৺ෆશʹͳΒͳ͍ͨΊʹɼೖӃ͠ͳ͍ํ͕ྑ͍Αʯͱ͸ݴ͑ͳ͍ɻ૬͍ؔͯ͠Δ͚ͩɻ
   9
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   :
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6. $BVTBMNPEFMWT1SFEJDUJWFNPEFMม਺બ୒
   q 1SFEJDUJWFNPEFMʹ͓͚Δม਺બ୒
   o ༧ଌೳྗΛߴΊΒΕΔม਺Ͱ͋Ε͹ԿͰ΋ྑ͍ͷͰɼࣗಈͷม਺બ୒ΞϧΰϦζϜΛ༻͍Δࣄ͕ଟ͍
   Ø -BTTP 3JEHFͳͲͷTISJOLBHFख๏
   Ø /FVSBMOFUXPSLʹΑΔख๏
   o ϞσϧͷධՁDSPTTWBMJEBUJPO
   o Ϟσϧ͕CMBDLCPY ˠม਺͕બ͹ΕͨܦҢ΍ΞϧΰϦζϜ্͕ख͘ػೳͨ͠ཧ༝ͳͲ͕ෆ໌
   Ø 8IBUJGஶऀ ʮͰ΋QSFEJDUJWFNPEFMͳΜ͔ʹղऍੑͳΜͯඞཁͳ͍ΑͶ ෆ໌Ͱ0,ʯ
   Ø Ϗδωεք۾ ʮΫϥΠΞϯτʹઆ໌͢ΔͨΊʹ͋Δఔ౓ͷղऍ͕ٻΊΒΕΔͷͰɼ
   USFFܥͷΞϧΰϦζϜ΍ͷ4)"1ͳͲ࢖͏ඞཁ͕͋Δɻʯ
   q $BVTBM BOBMZTJTʹ͓͚Δม਺બ୒
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   o ม਺બ୒ΞϧΰϦζϜΛ༻͍ΔͱɼਪఆྔʹόΠΞεΛੜΉՄೳੑ͕͋Δɻ
   ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
   

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7. ࠓ೔ͷίϯςϯπ
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   ҼՌਪ࿦ͱ༧ଌͷҧ͍
   
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   ௐ੔͢Δ͜ͱͰόΠΞεΛൃੜͤ͞Δม਺
   
   $BVTBMJOGFSFODFBOENBDIJOFMFBSOJOH
   ߴ࣍ݩͷڞมྔΛػցֶशͰௐ੔͢Δࡍͷ໰୊఺ͳͲ
   
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   %PVCMZSPCVTUFTUJNBUPSΛػցֶशͰߦ͏ख๏ɾςΫχοΫ
   
   7BSJBCMFTFMFDUJPOJTBEJGGJDVMUQSPCMFN
   ҼՌਪ࿦ͷม਺બ୒͸΍ͬͺΓ೉͍͠
   
   

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8. ͷαϚϦʔ
   ü ҼՌޮՌͷਪఆྔʹCJBTΛҾ͖ىͨ͜͠ΓɼCJBTΛ૿෯ͤͯ͞͠·͏ڞมྔ͕ଘࡏ͢Δ
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   ü #JBTΛҾ͖ى͜͢ڞมྔ͸σʔλ͔Β൑அͰ͖ͳ͍ͷͰɼFYQFSU LOPXMFEHF TVCKFDU
   NBUUFSLOPXMFEHFυϝΠϯ஌ࣝ
   Λ༻͍ͯDBVTBMHSBQIΛඳ͖ɼ൑அ͢Δඞཁ͕͋Δɻ
   ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
   

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9. ख࣋ͪͷશͯͷڞมྔΛௐ੔ʹ༻͍Δ΂͖͔
   ҼՌޮՌΛਪఆ͢Δࡍɼશͯͷڞมྔ9Λௐ੔͢Δ͜ͱ͸Մೳ͕ͩɼৗʹͦ͏͢Δ΂͖͔ʁ
   ͜Μͳঢ়گΛߟ͑ͯΈΔ
   q αϯϓϧαΠζ΍ม਺͕ແݶʹ͋Δ
   q Ϛγϯύϫʔ΋ແݶʹ͋Γɼ޷͖ͳ͚ͩڞมྔΛௐ੔Ͱ͖Δ
   • Ξ΢τΧϜ :
   
   • հೖม਺ "
   
   • ަབྷҼࢠ -
   ΛؚΉڞมྔ 9
   
   ౴͑͸/P
   ҼՌޮՌͷਪఆྔʹCJBTΛҾ͖ى͜͢ڞมྔ͕ଘࡏ͢Δ͔Β
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   1PTUUSFBUNFOUWBSJBCMF
   1SFUSFBUNFOUWBSJBCMF
   ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
   

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10. 1PTUUSFBUNFOUWBSJBCMFڞมྔʹDPMMJEFS͕͋Δ৔߹ 'JHVSFBOE
    
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    E[Y |A = 1] E[Y |A = 0]
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    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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11. 1PTUUSFBUNFOUWBSJBCMFڞมྔʹOPODPMMJEFS͕͋Δ৔߹ 'JHVSF
    
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    Ø -Λ৚݅෇͚Δ͜ͱͰTFMFDUJPOCJBT͕ੜ͡Δ
    b
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    ਅͷҼՌޮՌ͕ଘࡏ͠ͳ͍࣌ 'JHVSF`
    
    q "ͱ:͕ಠཱ ͔ͭ "ͱ-͕ಠཱ
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    =
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
    E[Y a=1] E[Y a=0]
    AAAB/nicdZDNSsNAFIUn9a/Wv6i4cjNYBDeGJBWrC6EogssKtlbSWCbTaTt0MgkzE6GEgK/ixoUibn0Od76N0zaCih4Y+Dj3Xu6dE8SMSmXbH0ZhZnZufqG4WFpaXlldM9c3mjJKBCYNHLFItAIkCaOcNBRVjLRiQVAYMHIdDM/G9es7IiSN+JUaxcQPUZ/THsVIaatjbp17N7cpOnEyfz9HO/M7Ztm2nGrFdo/hFA6dHFwHOpY9URnkqnfM93Y3wklIuMIMSek5dqz8FAlFMSNZqZ1IEiM8RH3iaeQoJNJPJ+dncFc7XdiLhH5cwYn7fSJFoZSjMNCdIVID+bs2Nv+qeYnqHfkp5XGiCMfTRb2EQRXBcRawSwXBio00ICyovhXiARIIK51YSYfw9VP4PzRdy6lY7uVBuXaax1EE22AH7AEHVEENXIA6aAAMUvAAnsCzcW88Gi/G67S1YOQzm+CHjLdP8weU2Q==
    = E[Y |A = 1] E[Y |A = 0]
    AAAB+3icdZDLSsNAFIYn9VbrLdalm8EiuDEkqVhdFKoiuKxgWyUNZTKdtEMnF2YmYol9FTcuFHHri7jzbZw2EVT0h4GP/5zDOfN7MaNCmuaHVpibX1hcKi6XVlbX1jf0zXJbRAnHpIUjFvFrDwnCaEhakkpGrmNOUOAx0vFGZ9N655ZwQaPwSo5j4gZoEFKfYiSV1dPL9XPn5v6kbrn7GZhuT6+YhlWrmvYxzODQysG2oGWYM1VArmZPf+/2I5wEJJSYISEcy4ylmyIuKWZkUuomgsQIj9CAOApDFBDhprPbJ3BXOX3oR1y9UMKZ+30iRYEQ48BTnQGSQ/G7NjX/qjmJ9I/clIZxIkmIs0V+wqCM4DQI2KecYMnGChDmVN0K8RBxhKWKq6RC+Pop/B/atmFVDfvyoNI4zeMogm2wA/aABWqgAS5AE7QABnfgATyBZ22iPWov2mvWWtDymS3wQ9rbJ9FrkwQ=
    = 0
    AAAB6XicdVBNS8NAEJ3Ur1q/qh69LBbBU0hSsXoQil48VrEf0Iay2W7apZtN2N0IJfQfePGgiFf/kTf/jds2goo+GHi8N8PMvCDhTGnH+bAKS8srq2vF9dLG5tb2Tnl3r6XiVBLaJDGPZSfAinImaFMzzWknkRRHAaftYHw189v3VCoWizs9Sagf4aFgISNYG+n2wumXK47t1qqOd44W5NTNieci13bmqECORr/83hvEJI2o0IRjpbquk2g/w1Izwum01EsVTTAZ4yHtGipwRJWfzS+doiOjDFAYS1NCo7n6fSLDkVKTKDCdEdYj9dubiX953VSHZ37GRJJqKshiUZhypGM0exsNmKRE84khmEhmbkVkhCUm2oRTMiF8fYr+Jy3Pdqu2d3NSqV/mcRThAA7hGFyoQR2uoQFNIBDCAzzBszW2Hq0X63XRWrDymX34AevtE0p7jTU=
    ਅͷҼՌޮՌ
    
    ৚݅෇͚ͳ͠
    "TTPDJBUJPONFBTVSF
    
    ৚݅෇͚͋Γ
    "TTPDJBUJPONFBTVSF
    
    ਅͷҼՌޮՌ͕ଘࡏ͢Δ࣌ͷΈੜ·ΕΔόΠΞε TFMFDUJPOCJBTVOEFSUIFBMUFSOBUJWF
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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12. 1PTUUSFBUNFOUWBSJBCMFڞมྔʹNFEJBUPS͕͋Δ৔߹ 'JHVSF
    
    q ޮՌͷϧʔτ͕ͭ͋Δ
    "͔Β:΁ͷ௚઀తͳޮՌ ͜Ε͕஌Γ͍ͨ
    
    -Λ௨ͨ͡"͔Β:΁ͷޮՌ
    q -Λ৚͚݅ͮͨ৔߹ɼਅͷҼՌޮՌͱ͸ҟͳͬͨਪఆྔ͕ಘΒΕΔ
    q ͳͥ͜͏ͳΔ͔ʁ
    Ø "͔Β:΁ͷޮՌ͕-ʹΑͬͯϒϩοΫ͞ΕΔ͔Β
    Ø 0WFSBEKVTUNFOUGPSNFEJBUPST ͱݺ͹ΕΔ
    E[Y a=1] E[Y a=0] 6=
    X
    l
    E[Y |L = l, A = 1]Pr[L = l]
    X
    l
    E[Y |L = l, A = 0]Pr[L = l]
    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
    ਅͷҼՌޮՌ -Ͱ৚͚݅ͮͨBTTPDJBUJPONFBTVSF
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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13. ͦ΋ͦ΋1PTUUSFBUNFOUWBSJBCMFΛௐ੔͠ͳ͚Ε͹͍͍ͷͰ͸ʁͱ͍͏ٙ໰
    ͜Ε·Ͱݟͨͭͷશͯͷྫʹ͓͍ͯʜ
    ü -͸ඞͣ"ͷޙʹ͘Δڞมྔ QPTUUSFBUNFOUWBSJBCMF
    ͩͬͨ
    ü 1PTUUSFBUNFOUWBSJBCMFͷௐ੔ʹΑΓTFMFDUJPOCJBT͕ൃੜ͍ͯͨ͠
    ૉ௚ͳਓʮ͡Ό͋1PTUUSFBUNFOUWBSJBCMFͷௐ੔͠ͳ͍ํ͕͍͍ͷͰ͸ʁ
    "ͱ-Ͳ͕ͬͪ࣌ܥྻతʹઌ͔ͳΜͯ؆୯ʹΘ͔ΔΑͶʁʯ
    ൱
    ަབྷҼࢠΛऔΓ͜΅͢Մೳੑ͕͋Δ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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14. 1PTUUSFBUNFOUWBSJBCMFͰ͔ͭௐ੔͢Δ΂͖ڞมྔͷྫ 'JHVSF
    
    -͸QPTUUSFBUNFOUWBSJBCMF͕ͩɼ
    CBDLEPPSQBUIΛด͡Δ໾໨ΛՌ͍ͨͯ͠Δɻ
    -Λௐ੔͠ͳ͍ͱCJBT͕ൃੜ͢Δ͕ɼ
    -Λௐ੔͢Ε͹VOCJBTFEʹਪఆͰ͖Δ
    "ͱ-ͷ࣌ܥྻతͳॱং͔ΒͰ͸ͳ͘ɼ
    ʮ-͕"ͷEFTDFOEBOU͔൱͔ ʯΛ൑அ͢Δ΂͖
    ͜ͷ৔߹ɼ-͸QPTUUSFBUNFOU WBSJBCMF͕ͩɼௐ੔͢Δ΂͖ڞมྔͰ͋Δ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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15. ʮ-͕"ͷEFTDFOEBOU͔൱͔ʯͲ͏൑அ͢Δ͔ʁ
    %BUB͔Β͸
    ʮ-͕"ͷEFTDFOEBOU͔൱͔ʯ΋ʮ-͕DPOGPVOEFS͔DPMMJEFS͔ʯ
    ൑அͰ͖ͳ͍
    4VCKFDUNBUUFSLOPXMFEHFΛ༻͍ͯ$BVTBMHSBQIॻ͍ͯ൑அ͢Δ͔͠ͳ͍
    ࣗಈͷม਺બ୒ΞϧΰϦζϜͰ-ΛܾఆͰ͖ͳ͍
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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16. 1SFUSFBUNFOUWBSJBCMF Λߟ͑Δ
    ͜ͷ໰୊ઃఆͰ͋Ε͹
    ʮશͯͷQSFUSFBUNFOUDPWBSJBUFTΛௐ੔͢Ε͹CJBTΛ࠷খʹͰ͖ΔʯͱҰൠʹݴΘΕ͍ͯΔɻ
    ͔͜͠͠Ε͸ਖ਼͍͠ओுͰ͸ͳ͍
    ൓ྫΛͭࣔ͢ɻ
    ͜Μͳঢ়گΛߟ͑ͯΈΔ
    q $BVTBMHSBQIͰ -ˠ"ˠ:ͱ͍͏ॱ -͕"ΑΓલʹ͋Δ1SFUSFBUNFOUWBSJBCMF
    
    q OQ αϯϓϧαΠζ͕ม਺ͷ਺ΑΓे෼ଟ͍
    
    Ø ҼՌޮՌͷਪఆྔͷ͹Β͖ͭ WBSJBODF
    ͸ແࢹͰ͖Δ
    Ø ߟྀ͢΂͖͸ਪఆྔͷCJBTͷΈ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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17. q 'JHVSF
    o ަབྷҼࢠ6
    ͕ଘࡏ͢Δ͕ɼ-ʹΑΓCBDLEPPSQBUI͕ด͡ΒΕ͍ͯΔ
    o #BDLEPPSQBUI͕ด͍ͯ͡ΔͷͰɼNBSHJOBMͳBTTPDJBUJPONFBTVSFΛ
    ༻͍ͯҼՌޮՌΛόΠΞεͳ͘ਪఆՄೳ
    o ٯʹ-Λ৚݅෇͚Δͱύε͕։͖ɼFYDIBOHFBCJMJUZ͕ຬͨ͞Εͣɼ
    CJBT͕ൃੜ͢Δ
    Ø .CJBTͱݺͿ .ͷܗΛͯ͠Δ͔ΒDIBQUFSࢀর
    
    o -͕DPMMJEFS͔DPOGPVOEFS͔σʔλ͔Β൑அͰ͖ͳ͍
    1SFUSFBUNFOUWBSJBCMF͕໰୊ʹͳΔέʔε.CJBT
    b
    'JHVSFͱಉ͡άϥϑ
    q 'JHVSF`-͔Β"ʹӨڹ͕͋Δ৔߹
    o -͸DPMMJEFSͰ΋DPOGPVOEFSͰ΋͋ΓɼҼՌޮՌ͸ਪఆෆՄೳ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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18. q ໰୊ઃఆ
    o ະଌఆͷަབྷҼࢠ6͕ଘࡏ
    o "ʹͷΈӨڹΛ༩͑Δม਺;͕ଘࡏ ަབྷҼࢠͰ͸ͳ͍
    
    Ø ;͸ૢ࡞ม਺ͱݺ͹ΕΔ DIBQUFSࢀর
    
    q ;ͷௐ੔ͳ͋͠ΓͰҼՌޮՌΛਪఆ͢Δͱʜ
    Ø ;ௐ੔ͳ͠ਪఆʹΑΔόΠΞεΑΓɼ;ௐ੔͋ΓਪఆʹΑΔόΠΞε͕
    େ͖͘ͳͬͨΓ BNQMJGZ
    ɼখ͘͞ͳͬͨΓ SFEVDF
    ͢Δ
    Ø ;CJBTͱݺ͹ΕɼͲͬͪ BNQMJGZPSSFEVDF
    ʹಇ͔͘͸Θ͔Βͳ͍
    Ø ૢ࡞ม਺͸ௐ੔ͯ͠͸μϝ
    1SFUSFBUNFOUWBSJBCMF͕໰୊ʹͳΔέʔε;CJBT CJBTBNQMJGJDBUJPO
    
    'JHVSFͱಉ͡άϥϑ
    ;Λ৚݅෇͚ͣਪఆ͠
    ى͜ΔόΠΞε
    (E[Y a=1] E[Y a=0]) (E[Y |A = 1] E[Y |A = 0])
    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
    ਅͷҼՌޮՌ ৚݅෇ͳ͠ͷਪఆྔ
    ʹ
    (E[Y a=1] E[Y a=0]) (
    X
    l
    E[Y |Z = z, A = 1]Pr[Z = z]
    X
    l
    E[Y |Z = z, A = 0]Pr[Z = z])
    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
    ;Λ৚݅෇͚ͯਪఆ͠
    ى͜ΔόΠΞε
    ਅͷҼՌޮՌ
    ;Λ৚݅෇͚ͨͱ͖ͷਪఆྔ
    ʹ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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19. ͷαϚϦʔ
    ü ҼՌޮՌͷਪఆ஋ʹCJBTΛҾ͖ىͨ͜͠ΓɼCJBTΛ૿෯ͤͯ͞͠·͏ڞมྔ͕ଘࡏ͢Δ
    ͷͰɼҋӢʹԿͰ΋͔ΜͰ΋ڞมྔΛௐ੔͢Δ΂͖Ͱ͸ͳ͍ɻ
    ü #JBTΛҾ͖ى͜͢ڞมྔ͸σʔλ͔Β൑அͰ͖ͳ͍ͷͰɼFYQFSU LOPXMFEHF TVCKFDU
    NBUUFSLOPXMFEHFυϝΠϯ஌ࣝ
    Λ༻͍ͯDBVTBMHSBQIΛඳ͖ɼ൑அ͢Δඞཁ͕͋Δɻ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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20. ࠓ೔ͷίϯςϯπ
    5IFEJGGFSFOUHPBMTPGWBSJBCMFTFMFDUJPO
    ҼՌਪ࿦ͱ༧ଌͷҧ͍
    
    7BSJBCMFTUIBUJOEVDFPSBNQMJGZCJBT
    ௐ੔͢Δ͜ͱͰόΠΞεΛൃੜͤ͞Δม਺
    
    $BVTBMJOGFSFODFBOENBDIJOFMFBSOJOH
    ߴ࣍ݩͷڞมྔΛػցֶशͰௐ੔͢Δࡍͷ໰୊఺ͳͲ
    
    %PVCMZSPCVTUNBDIJOFMFBSOJOHFTUJNBUPST
    %PVCMZSPCVTUFTUJNBUPSΛػցֶशͰߦ͏ख๏ɾςΫχοΫ
    
    7BSJBCMFTFMFDUJPOJTBEJGGJDVMUQSPCMFN
    ҼՌਪ࿦ͷม਺બ୒͸΍ͬͺΓ೉͍͠
    
    

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21. ͷαϚϦʔ
    ü #JBTΛҾ͖ى͜͢ม਺ JOEVDFPSBNQMJGZCJBT
    Λؚ·ͳ͍ߴ࣍ݩͷσʔλ͕͋Δͱ͖ɼ
    ަབྷҼࢠͷௐ੔ʹػցֶशΛ࢖͑Δ͚Ͳɼ৭ʑͱ໰୊΋͋Δɻ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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22. ͜Μͳঢ়گΛߟ͑ͯΈΔ
    q "ͱ:ͷଞʹɼେྔͷڞมྔ9ΛؚΜͩσʔλ ߴ࣍ݩσʔλ
    
    Ø &YQFSULOPXMFEHFΛ༻͍ͯɼ9͔ΒʮόΠΞεΛҾ͖ى͜͢ม਺ʯΛऔΓআ͍͍ͯΔ
    Ø 9DPOGPVOEFSOPODPOGPVOEFS
    q αϯϓϧαΠζ /
    ڞมྔͷݸ਺ 1
    ͷঢ়گ
    Ø /1͕ͩ1͕ඇৗʹ/ʹ͍ۙ
    Ø ΋͘͠͸ 1/Ͱɼڞมྔͷ਺ͷํ͕ଟ͍
    ߴ࣍ݩσʔλͷ໰୊ઃఆ
    ͜ͷͱ͖ɼௐ੔͢ΔڞมྔΛͲͷΑ͏ʹબΜͰҼՌޮՌΛਪఆ͢Δ͔ʁ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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23. छྨͷڞมྔͷௐ੔ํ๏
    Λਪఆ
    
    HGPSNVMB
    9Λ৚݅෇͚ͨͱ͖ͷ:ͷฏۉ
    
    *1XFJHIUJOH
    9Λ৚͚݅ͮͨͱ͖ͷ"ͷ֬཰
    b(X) = E[Y |A = 1, L]
    AAAB+XicdVDJSgNBEO2JW4zbqEcvjUGIIGEmm8khEBXBg4cIZpFkCD2dnqRJz0J3TyCM+RMvHhTx6p9482/sSSKo6IOCx3tVVNWzA0aFNIwPLbG0vLK6llxPbWxube/ou3tN4Ycckwb2mc/bNhKEUY80JJWMtANOkGsz0rJHF7HfGhMuqO/dyklALBcNPOpQjKSSerpuZ9rH1cvO3f1Z1TyB11ZPTxvZXL6cL1ZgTEqlgjEjlVLRhGbWmCENFqj39Pdu38ehSzyJGRKiYxqBtCLEJcWMTFPdUJAA4REakI6iHnKJsKLZ5VN4pJQ+dHyuypNwpn6fiJArxMS1VaeL5FD89mLxL68TSqdsRdQLQkk8PF/khAxKH8YxwD7lBEs2UQRhTtWtEA8RR1iqsFIqhK9P4f+kmcua+WzuppCunS/iSIIDcAgywASnoAauQB00AAZj8ACewLMWaY/ai/Y6b01oi5l98APa2ycIV5H9
    ⇡(L) = Pr(A = 1|L)
    AAAB+3icdVDLSgMxFM3UV62vsS7dBIvQbspMX7aLQtWNiy4q2Ae0pWTSTBuayQxJRixjf8WNC0Xc+iPu/BvTh6CiBy4czrmXe+9xAkalsqwPI7a2vrG5Fd9O7Ozu7R+Yh8mW9EOBSRP7zBcdB0nCKCdNRRUjnUAQ5DmMtJ3J5dxv3xIhqc9v1DQgfQ+NOHUpRkpLAzPZC2i6noFV2BDp86p9X88MzJSVzeXL+WIFzkmpVLAWpFIq2tDOWgukwAqNgfneG/o49AhXmCEpu7YVqH6EhKKYkVmiF0oSIDxBI9LVlCOPyH60uH0GT7UyhK4vdHEFF+r3iQh5Uk49R3d6SI3lb28u/uV1Q+WW+xHlQagIx8tFbsig8uE8CDikgmDFppogLKi+FeIxEggrHVdCh/D1KfyftHJZO5/NXRdStYtVHHFwDE5AGtjgDNTAFWiAJsDgDjyAJ/BszIxH48V4XbbGjNXMEfgB4+0TEbiSiQ==
    Λਪఆ
    C 9
    ͱК 9
    ͸ڞʹઢܗճؼ 0-4
    ΍ϩδεςΟοΫճؼͰਪఆͰ͖Δɻ
    ਪఆͷࡍʹ͸ަ߲ࠩ΍DVCJDTQMJOFͳͲͰ͖Δ͚ͩଟ͘ͷ৘ใΛ੝ΓࠐΜͰɼਅͷϞσϧʹ͚͍ۙͮͨɻ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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24. ճؼͰߴ࣍ݩσʔλΛѻ͏ࡍͷ໰୊఺
    q /1 ͚ͩͲ1͕୔ࢁ͋Δ৔߹
    o ઢܗճؼ΍ϩδεςΟοΫճؼͰύϥϝʔλͷղ͸ٻ·Δ͕ɼਪఆ஋ͷ෼ࢄ͕େ͖͘ෆ҆ఆʹͳΔ
    Ø C 9
    ΍К 9
    ͷਪఆ஋΋ෆ҆ఆʹͳΔɻ
    q 1/ ڞมྔͷ਺ͷํ͕ଟ͍
    ͷ৔߹
    o ճؼʹ͓͍ͯύϥϝʔλͷղ͕Ұҙʹఆ·Βͳ͍ɻղ͕ແݶʹଘࡏ͢ΔͷͰਪఆෆՄೳɻ
    Ø Ќ 9`9
    9`:ͷ9`9͕ඇਖ਼ଇߦྻͱͳΓٯߦྻΛܭࢉͰ͖ͳ͍ͨΊ
    Ø ৄ͘͠͸ͪ͜ΒIUUQXXXFCTUBOGPSEFEVdIBTUJF1BQFSTQHUOQEG
    q К 9
    ͷਪఆʹ͓͍ͯɼ"ͱڧ͍૬ؔΛ΋ͭOPODPOGPVOEFSΛϞσϧʹೖΕΔͱQPTJUJWJUZ͕ຬͨ͞Εͳ͘ͳΔ
    Ø ෼ࢄരൃ DIBQUFSࢀর
    
    ػցֶशͰղܾ͠Α͏ʂ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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25. ҼՌਪ࿦Ͱߴ࣍ݩσʔλΛѻ͏ػցֶशख๏େ͖͘छྨ
    1SFEJDUJPO༻ͷม਺બ୒ΞϧΰϦζϜ
    o 'PSXBSETUFQXJTFఆ਺߲ͷΈͷϞσϧ͔Β࢝Ίͯɼͭͣͭม਺Λ଍͍͖ͯ͠ͳ͕Βਫ਼౓ΛධՁ
    o CBDLXBSETUFQXJTFϑϧϞσϧ͔Β࢝Ίͯɼͭͣͭม਺Λআ͍͍͖ͯͳ͕Βਫ਼౓ΛධՁ
    4ISJOLBHFख๏ ਖ਼ଇԽख๏
    
    o -BTTPճؼଛࣦؔ਺ʹ-ϊϧϜͷQFOBMUZΛ՝ͯ͠࠷దԽΛߦ͏
    Ø ༧ଌʹ໾ཱͨͳ͍ڞมྔͷύϥϝʔλͷਪఆ஋͕ʹͳΔ
    ߴ࣍ݩΛѻ͑ΔඇઢܗͳػցֶशΞϧΰϦζϜ
    o 5SFFܥ3BOEPN'PSFTUɼ9HCPPTUɼ-JHIU(#.
    Ø Ϟσϧ಺ͰΞϯαϯϒϧֶश ଟ਺ܾ
    ߦ͏ͨΊɼ༧ଌ஋ͷόϦΞϯεΛ௿͘཈͑ΒΕΔɻ
    Ø ֶशεϐʔυ͕଎͍͠ɼ༧ଌਫ਼౓΋ྑ͍ɻ
    o /FVSBM/FUXPSL%FFQMFBSOJOH
    Ø %SPQPVUͱ͍͏ਖ਼ଇԽʹࣅͨςΫχοΫΛ࢖͏͜ͱͰɼաֶशΛىͣ͜͞༧ଌਫ਼౓ΛߴΊΒΕΔɻ
    Ø Ϛγϯύϫʔ͕ඞཁͰɼ͔ͭܭࢉ͕ͱʹ͔͘஗͍ɻ
    ม਺ΛݮΒͦ͏
    ܎਺Λʹ͠Α͏
    ΞϧΰϦζϜʹ
    ೚ͤΑ͏
    Φεεϝͷػցֶशຊ
    IUUQGBDVMUZNBSTIBMMVTDFEVH
    BSFUIKBNFT*4-
    ͔͠͠ɼେ͖ͳ໰୊͕఺͋Δ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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26. ҼՌਪ࿦ͷަབྷௐ੔ʹ͓͍ͯػցֶशΛ༻͍Δ͜ͱͷ໰୊఺ͭ
    બ୒͞Εͨม਺͕ަབྷΛऔΓআ͍͍ͯΔͱ͸ݶΒͳ͍
    o 9OPODPOGPVOEFSDPOGPVOEFSͳͷͰɼແବͳม਺ΛؚΜͰަབྷҼࢠΛऔΓ͜΅͍ͯ͠ΔՄೳੑ͋Γ
    o ϞσϧͷNJTTQFTJGJDBUJPO͕ى͍ͬͯ͜Δ৔߹ɼਪఆ஋͕όΠΞεΛؚΜͰ͠·͏
    o ͜ͷ໰୊͸%PVCMZ3PCVTU %3
    FTUJNBUPSͰվળͰ͖ΔՄೳੑ͕͋Δ
    ਪఆ஋ͷ৴པ͕۠ؒখ͘͞ͳΔՄೳੑ
    o %3FTUJNBUPSΛ༻͍ͯVOCJBTFEͳਪఆ஋Λಘͨͱͯ͠΋ɼਪఆ஋ͷ෼ࢄ͕ޡͬͯਪఆ͞ΕΔՄೳੑ͕͋Δɻ
    o ͜ͷਪఆ஋ͷ෼ࢄΛ༻͍ͯ৴པ۠ؒΛਪఆ͢Ε͹ɼ৴པ۠ؒͷैདྷͷղऍ͸Ͱ͖ͳ͘ͳΔɻ
    C 9
    ͱК 9
    Λ૊Έ߹Θͤͯਪఆ஋ͷόΠΞεΛখ͘͢͞Δํ๏
    ฼ूஂ͔ΒαϯϓϦϯάΛ܁Γฦ͠ਪఆͨ͠৔߹ɼ
    ͦͷ͏ͪ͸͜ͷൣғʹதʹೖΔ
    ͜ΕΒͭͷ໰୊ΛղܾͰ͖ΔΑ͏ʹ
    %PVCMZSPCVTUFTUJNBUPSΛػցֶश༻ʹվྑ͠Α͏
    ࢲͷٙ໰
    ʮͳΜͰʁ
    0WFSGJUUJOH ʯ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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27. ࠓ೔ͷίϯςϯπ
    5IFEJGGFSFOUHPBMTPGWBSJBCMFTFMFDUJPO
    ҼՌਪ࿦ͱ༧ଌͷҧ͍
    
    7BSJBCMFTUIBUJOEVDFPSBNQMJGZCJBT
    ௐ੔͢Δ͜ͱͰόΠΞεΛൃੜͤ͞Δม਺
    
    $BVTBMJOGFSFODFBOENBDIJOFMFBSOJOH
    ߴ࣍ݩͷڞมྔΛػցֶशͰௐ੔͢Δࡍͷ໰୊఺ͳͲ
    
    %PVCMZSPCVTUNBDIJOFMFBSOJOHFTUJNBUPST
    %PVCMZSPCVTUFTUJNBUPSΛػցֶशͰߦ͏ख๏ɾςΫχοΫ
    
    7BSJBCMFTFMFDUJPOJTBEJGGJDVMUQSPCMFN
    ҼՌਪ࿦ͷม਺બ୒͸΍ͬͺΓ೉͍͠
    
    

    View Slide

28. %PVCMZSPCVTU %3
    FTUJNBUPSͬͯͳΜ͚ͩͬ $IBQUFSͷ෮श
    
    q %PVCMZSPCVTU %3
    FTUJNBUPS
    o *1XFJHIUJOHͱTUBOEBSEJ[BUJPO QMVHJOHGPSNVMB
    ͷਪఆྔΛ૊Έ߹Θͤͨਪఆྔ
    o ؆୯ͷͨΊҼՌޮՌͰ͸ͳ͘ɼ&<:B>DPVOUFSGBDUVSBMNFBOPVUDPNFVOEFSUSFBUNFOUΛߟ͑Δ
    Ø *1XFJHIUJOHͱTUBOEBSEJ[BUJPOͦΕͧΕͷFTUJNBOE ਅ஋
    ͸
    Ø ͦΕͧΕͷFTUJNBUPS ਪఆྔ
    ͸
    o %3FTUJNBUPS͸
    ˆ
    E[Y a=1] =
    8
    >
    >
    >
    >
    >
    <
    >
    >
    >
    >
    >
    :
    1
    n
    n
    X
    i=1
    ˆ
    b(Li)
    1
    n
    n
    X
    i=1
    AiYi
    ˆ
    ⇡(Li)
    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
    E[Y a=1] =
    8
    <
    :
    E[b(L)]
    E[ AY
    ⇡(L)
    ]
    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
    if IP weighting, where ⇡(L) = Pr(A = 1|L)
    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
    if standardization, where b(L) = E[Y |A = 1, L]
    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
    ˆ
    E[Y a=1]DR =
    1
    n
    n
    X
    i=1
    {ˆ
    b(Li) +
    Ai
    ˆ
    ⇡(Li)
    (Yi
    ˆ
    b(Li))}
    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
    !
    !
    ͸JOEJWJEVBM͝ͱͷ:ͷ༧ଌ஋
    $
    π !
    ͸JOEJWJEVBM͝ͱͷ"ͷ༧ଌ஋
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

    View Slide

29. %PVCMZSPCVTU %3
    FTUJNBUPSͬͯͳΜ͚ͩͬ $IBQUFSͷ෮श
    
    o %PVCMZSPCVTU %3
    FTUJNBUPS
    o %3FTUJNBUPS͸
    o %3 FTUJNBUPSͷόΠΞε͸େඪຊͷͱ͖
    Ø όΠΞεʹ͸ !
    "($)
    − !
    "∗($)
    
    ͱ − ∗()
    ͷੵؚ͕·Ε͍ͯΔ
    Ø C -
    ͱК -
    ͷͲͪΒ͔ͷϞσϧ͕ਖ਼͚͠Ε͹ɼόΠΞε͸ʹͳΔ
    ˆ
    E[Y a=1]DR =
    1
    n
    n
    X
    i=1
    {ˆ
    b(Li) +
    Ai
    ˆ
    ⇡(Li)
    (Yi
    ˆ
    b(Li))}
    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
    E[ ˆ
    E[Y a=1]DR E[Y a=1]] = E[⇡(L)(
    1
    ⇡(L)
    1
    ⇡⇤(L)
    )(b(L) b⇤(L))]
    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
    К -
    К -
    JGUSFBUNFOUNPEFM JTDPSSFDU
    C -
    C -
    JGPVUDPNFNPEFMJTDPSSFDU
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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30. o ػցֶशͱ%3FTUJNBUPSΛ༻͍ͯ෼ੳΛߦͬͨͱͯ͠ɼҎԼΛ૝ఆ
    ü ਪఆྔͷόΠΞεΛඇৗʹখ͘͞Ͱ͖ͨ !
    √(
    ΑΓ΋খ͘͞
    
    ü ਪఆྔͷόΠΞε ਪఆྔͷ෼ࢄͷόΠΞε
    %3FTUJNBUPSͷཧ૝తͳར༻
    ͨͱ͑΋͠C -
    ͱК -
    ͷͦΕͧΕͷਪఆྔͷόΠΞε͕ !
    √(
    ΑΓେ͖ͯ͘΋ɼ
    %3 FTUJNBUPSͷόΠΞε͸খ͘͞ͳΓಘΔ !
    "($)
    − !
    "∗($)
    
    ͱ − ∗()
    ͷੵ͔ͩΒɻ
    ͜ͷόΠΞεͷখ͍͞ਪఆྔΛ༻͍Ε͹ɼWBMJEͳ৴པ۠ؒΛܭࢉͰ͖Δ͸ͣɻ
    E[ ˆ
    E[Y a=1]DR E[Y a=1]] = E[⇡(L)(
    1
    ⇡(L)
    1
    ⇡⇤(L)
    )(b(L) b⇤(L))]
    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
    o σʔλ͕େن໛ඪຊͷ৔߹
    ü ʮҰகਪఆྔʯ͔ͭʮਅ஋Λฏۉʹ࣋ͭਖ਼ن෼෍ʹै͏ʯΑ͏ͳ%3FTUJNBUPSΛఆٛͰ͖Δɻ
    ü ͜ͷਖ਼ن෼෍Λ༻͍ͯɼ৴པ۠ؒΛܭࢉͰ͖ͦ͏ɻ
    E[ ˆ
    E[Y a=1]DR E[Y a=1]] <
    1
    p
    n
    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
    ͭͷςΫχοΫɼTQMJUUJOHͱDSPTTGJUUJOHΛ%3FTUJNBUPSʹಋೖ͠Α͏
    ࢲͷٙ໰
    ʮ෼ࢄ͕খ͘͞ͳΔ࿩ͱ
    Ͳ͏ܨ͕Δͷ ʯ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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31. 4UVEZ
    QPQVMBUJPO
    %3FTUJNBUPSͱػցֶशͷ૊Έ߹ΘͤʹඞཁͳςΫχοΫTQMJUUJOH
    &TUJNBUJPO
    TBNQMF
    5SBJOJOH
    TBNQMF
    !
    ()
    &
    ()
    !
    ()
    &
    π()
    ATEDR = ˆ
    E[Y a=1]DR
    ˆ
    E[Y a=0]DR
    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
    શαϯϓϧ 4UVEZQPQVMBUJPO
    Λ෼ׂ͢Δ
    o 5SBJOJOHTBNQMFͱ &TUJNBUJPOTBNQMFͱݺͿ
    5SBJOJOHTBNQMFΛ༻͍ͯ&
    ()ͱ)
    ()ͷϞσϧΛ࡞੒͢Δɻ
    o ͜ͷͭ͸ֶशࡁΈϞσϧͱݟͳͤΔɻ
    ֶशࡁΈϞσϧΛ༻͍ͯ&TUJNBUJPOTBNQMFͰͷCͱКͷਪఆ஋
    &
    ()ͱ)
    π()Λਪఆ͢Δɻ
    &
    ()ͱ)
    π()Λ༻͍ͯ%3ฏۉҼՌޮՌ "5&%3
    
    Λਪఆ͢Δɻ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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32. %3FTUJNBUPSͱػցֶशͷ૊Έ߹ΘͤʹඞཁͳςΫχοΫDSPTTGJUUJOH
    4UVEZ
    QPQVMBUJPO
    &TUJNBUJPO
    TBNQMF
    5SBJOJOH
    TBNQMF
    "
    ()
    '
    ()
    "
    ()
    '
    π()
    ATEDR = ˆ
    E[Y a=1]DR
    ˆ
    E[Y a=0]DR
    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
    &TUJNBUJPO
    TBNQMF
    5SBJOJOH
    TBNQMF
    "
    ()
    '
    ()
    "
    ()
    '
    π()
    ATEDR = ˆ
    E[Y a=1]DR
    ˆ
    E[Y a=0]DR
    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
    4QMJUUJOHͷσϝϦοτ
    ҼՌޮՌͷਪఆͷࡍʹ൒෼ͷσʔλΛࣺͯΔ͜ͱʹͳΓɼ΋͍ͬͨͳ͍ɻ
    શαϯϓϧ࢖ͬͨਪఆ஋ͷ৴པ۠ؒΑΓ޿͘ͳͬͨΓͦ͠͏ɻ
    $SPTTGJUUJOHͷΞΠσΞ
    4QMJUUJOHΛճߦ͏͜ͱͰ5SBJOJOHͱ&TUJNBUJPOͷ໾ׂΛεΠον͠ɼ
    ͭͷฏۉҼՌޮՌΛग़ͦ͏
    ࠷ऴతͳશαϯϓϧͰͷฏۉҼՌޮՌ͸ͭͷҼՌޮՌͷฏۉΛऔͬͯಘΔɻ
    4QMJU
    4QMJU
    Entire ATEDR =
    AAACAHicdVDLSsNAFJ3UV62vqAsXbgaL4CokragboT4KLqu0tdCGMJlO2qGTSZiZiCVk46+4caGIWz/DnX/j9CH4PDBwOOde7pzjx4xKZdvvRm5mdm5+Ib9YWFpeWV0z1zeaMkoEJg0csUi0fCQJo5w0FFWMtGJBUOgzcu0Pzkb+9Q0Rkka8roYxcUPU4zSgGCkteeZWJ/Sj27TKFRUEntSrmZeeX2XHnlm0rfKBrQF/E8eyxyiCKWqe+dbpRjgJCVeYISnbjh0rN0VCUcxIVugkksQID1CPtDXlKCTSTccBMrirlS4MIqEfV3Csft1IUSjlMPT1ZIhUX/70RuJfXjtRwZGbUh4ninA8ORQkDKoIjtqAXZ0aKzbUBGFB9V8h7iOBsNKdFXQJn0nh/6RZspyyVbrcL1ZOp3XkwTbYAXvAAYegAi5ADTQABhm4B4/gybgzHoxn42UymjOmO5vgG4zXD2P3lkU=
    ATEDR from split 2
    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
    ATEDR from split 1
    AAACDXicdVDLSgMxFM3Ud32NunQTrIKrYaZVOt3VF7hUsVZoh5JJMzaYTIYkI5ZhfsCNv+LGhSJu3bvzb0wfgooeCJyccy/33hMmjCrtuh9WYWJyanpmdq44v7C4tGyvrF4okUpMGlgwIS9DpAijMWloqhm5TCRBPGSkGV4fDPzmDZGKivhc9xMScHQV04hipI3UsTfbPBS32d75Ud7JDs9yOPrDSAoOldlAQy/v2CXXqe5U/KoPXcf1a65XHpBqZbdcg57jDlECY5x07Pd2V+CUk1hjhpRqeW6igwxJTTEjebGdKpIgfI2uSMvQGHGigmx4TQ63jNKFkZDmxRoO1e8dGeJK9XloKjnSPfXbG4h/ea1UR36Q0ThJNYnxaFCUMqgFHEQDu1QSrFnfEIQlNbtC3EMSYW0CLJoQvi6F/5OLsuNVnPLpTqm+P45jFqyDDbANPFAFdXAMTkADYHAHHsATeLburUfrxXodlRascc8a+AHr7RN8rpvW
    +
    AAAB6HicdVDJSgNBEK2JW4xb1KOXxiAIwjCTBOIx6MVjAiYGkiH0dGqSNj0L3T1CGPIFXjwo4tVP8ubf2FkE1wcFj/eqqKrnJ4Ir7TjvVm5ldW19I79Z2Nre2d0r7h+0VZxKhi0Wi1h2fKpQ8AhbmmuBnUQiDX2BN/74cubf3KFUPI6u9SRBL6TDiAecUW2k5lm/WHLsSrXmuBXym7i2M0cJlmj0i2+9QczSECPNBFWq6zqJ9jIqNWcCp4VeqjChbEyH2DU0oiEqL5sfOiUnRhmQIJamIk3m6teJjIZKTULfdIZUj9RPbyb+5XVTHZx7GY+SVGPEFouCVBAdk9nXZMAlMi0mhlAmubmVsBGVlGmTTcGE8Pkp+Z+0y7ZbscvNaql+sYwjD0dwDKfgQg3qcAUNaAEDhHt4hCfr1nqwnq2XRWvOWs4cwjdYrx/BEozp
    2
    AAAB6HicdVDJSgNBEK2JW4xb1KOXxiB4GmaSQDwGvXhMwCyQDKGnU5O06Vno7hHCkC/w4kERr36SN//GziK4Pih4vFdFVT0/EVxpx3m3cmvrG5tb+e3Czu7e/kHx8Kit4lQybLFYxLLrU4WCR9jSXAvsJhJp6Avs+JOrud+5Q6l4HN3oaYJeSEcRDzij2kjN8qBYcuxKtea4FfKbuLazQAlWaAyKb/1hzNIQI80EVarnOon2Mio1ZwJnhX6qMKFsQkfYMzSiISovWxw6I2dGGZIglqYiTRbq14mMhkpNQ990hlSP1U9vLv7l9VIdXHgZj5JUY8SWi4JUEB2T+ddkyCUyLaaGUCa5uZWwMZWUaZNNwYTw+Sn5n7TLtluxy81qqX65iiMPJ3AK5+BCDepwDQ1oAQOEe3iEJ+vWerCerZdla85azRzDN1ivH8uujPA=
    ৴པ۠ؒ͸#PPUTUSBQͰܭࢉ͢Δ
    GPME DSPTTWBMJEBUJPOͱ
    ಉ͡ΞΠσΞ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

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33. ͷαϚϦʔ
    ü ߴ࣍ݩσʔλ͔ΒػցֶशΛ༻͍ͯަབྷҼࢠͷબ୒Λߦ͏৔߹ʹ͸ɼϞσϧͷ
    NJTTQFDJGJDBUJPOͷՄೳੑ͕͋Δɻ
    ü ·ͨɼ৴པ͕۠ؒڱ͘ͳΔ໰୊΋ى͜Γ͏Δɻ
    ü 4QMJUUJOHͱDSPTTGJUUJOHΛ൐ͬͨ%PVCMZ3PCVTU&TUJNBUPSΛ༻͍Ε͹ɼNJTTQFDJGJDBUJPO
    ͷϦεΫΛݮΒͤΔ͠ɼWBMJEͳ৴པ۠ؒ΋ಘΔ͜ͱ͕Ͱ͖Δɻ
    ࢲͷٙ໰
    %3ਪఆྔΛ࢖Θͣʹɼ
    ී௨ʹHGPSNVMBΛ࢖͏৔߹ʹ΋DSPTTGJUUJOHͨ͠ํ͕
    WBMJEͳ৴པ۠ؒΛಘΒΕͨΓ͠ͳ͍ͷʁ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

    View Slide

34. ࠓ೔ͷίϯςϯπ
    5IFEJGGFSFOUHPBMTPGWBSJBCMFTFMFDUJPO
    ҼՌਪ࿦ͱ༧ଌͷҧ͍
    
    7BSJBCMFTUIBUJOEVDFPSBNQMJGZCJBT
    ௐ੔͢Δ͜ͱͰόΠΞεΛൃੜͤ͞Δม਺
    
    $BVTBMJOGFSFODFBOENBDIJOFMFBSOJOH
    ߴ࣍ݩͷڞมྔΛػցֶशͰௐ੔͢Δࡍͷ໰୊఺ͳͲ
    
    %PVCMZSPCVTUNBDIJOFMFBSOJOHFTUJNBUPST
    %PVCMZSPCVTUFTUJNBUPSΛػցֶशͰߦ͏ख๏ɾςΫχοΫ
    
    7BSJBCMFTFMFDUJPOJTBEJGGJDVMUQSPCMFN
    ҼՌਪ࿦ͷม਺બ୒͸΍ͬͺΓ೉͍͠
    
    

    View Slide

35. ෼ࢄരൃͷ໰୊
    o ཧ࿦తʹ͸ʮ%3ਪఆྔͷ෼ࢄʯʮC 9
    ΍К 9
    ͕ط஌ͷ৔߹ʹಘΒΕΔਪఆྔͷ෼ࢄʯͱͳΔ͸ͣɻ
    o ͔͠͠ɼ෼ࢄരൃ͕ى͍ͬͯ͜ΔՄೳੑ΋͋Γɼਪఆ஋ͷ৴པ͕۠ؒҙຯͷͳ͍΋ͷʹͳ͍ͬͯΔՄೳੑ͕͋Δɻ
    ࠷ڧͬΆ͍%3FTUJNBUPSػցֶशʹ΋৭ʑݶք໰୊఺͕͋Δ
    4VCKFDUNBUUFSLOPXMFEHFͷݶք
    o ઐ໳஌ࣝΛ༻͍ͯόΠΞεΛى͜͢ม਺ΛϐοΫΞοϓͯ͠΋ɼऔΓ͜΅͢ՄೳੑͳͲ͕े෼͋Δɻ
    o ΋ͦ͠͏ͳΔͱɼ%3FTUJNBUPS͸্ख͘ػೳ͠ͳ͍ɻ
    ΞϧΰϦζϜ΋DBVTBMTUSVDUVSF΋Θ͔ΒΜ໰୊
    o ʮͲΜͳ໰୊ʹͰ΋ສೳͳΞϧΰϦζϜʯ͸ଘࡏ͠ͳ͍ ϊʔϑϦʔϥϯνఆཧ
    
    o ΞϧΰϦζϜ͸DBVTBMTUSVDUVSFʹΑܾͬͯఆ͞ΕΔ΂͖͕ͩɼDBVTBMTUSVDUVSF΋ະ஌Ͱ͋Δɻ
    %3FTUJNBUPSػցֶश͸Ϛγϯύϫʔతʹݫ͍͠
    o UJNFWBSZJOHUSFBUNFOUT͕͋Δ෼ੳɼಛʹTVSWJWBMBOBMZTJTͳͲΛѻ͏ͷ͸ݱ࣮తͰ͸ͳ͍
    o ͜ͷล͸1BSUͰ΍Δͦ͏Ͱ͢
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

    View Slide

36. ෼ࢄരൃͷ໰୊
    o ෼ࢄരൃ
    Ø "ʹڧ͍૬ؔͷ͋Δڞมྔؚ͕·Ε͍ͯΔͱɼ෼ੳखॱ͕ਖ਼͔ͬͨ͠ͱͯ͠΋ಛఆͷ஋Λ࣋ͭਓͷК 9
    ͕΍ʹͳΔ
    Ø ͷ෼ࢄ͕ඇৗʹେ͖͘ͳΔ ෼฼͕ͷਓͱ͔͍Δ͔Β
    
    Ø ͦΕʹ൐ͬͯ৴པ۠ؒ΋޿͘ͳΔ
    $IBQUFSͷ,PP !NFEJ@EBUB
    ͞ΜൃදࢿྉΑΓεΫγϣ
    ˆ
    E[Y a=1] =
    1
    n
    n
    X
    i=1
    AiYi
    ˆ
    ⇡(Li)
    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
    ʮ͜Μͳ݁Ռ͓͔͍͠ʂ෼ࢄരൃΛى͜͢ม਺Λআ͍ͯ෼ੳΛ΍Γ௚ͧ͢ʂʯͱࢥͬͯ͠·͏͕ɼ
    ෼ੳલʹม਺Λዞҙతʹ࡟আ͢Ε͹ɼ࠷ऴతʹWBMJEͳ৴པ۠ؒΛಘΒΕΔอূ͕ͳ͘ͳΔɻ
    Ұମզʑ͸Ͳ͏͢Ε͹͍͍ͷ͔ʁ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

    View Slide

37. ࠷ޙʹXIBUJGஶऀ͔ΒͷΞυόΠε
    ҼՌਪ࿦ʹ͓͚ΔҰൠతͳม਺બ୒ͷํ๏࿦Λཱ֬͢Δͷ͸ͨͿΜແཧ ɻ
    ͨͩɼͭΞυόΠε͢ΔͳΒɼײ౓෼ੳ͸ͨ͠ํ͕͍͍ɻ
    ৭Μͳ෼ੳํ๏Λࢼͯ͠ൺֱͯ͠ΈΔɻ
    ΋݁͠Ռ͕Ұகͯ͠Ε͹خ͍͠ɻҰகͯ͠ͳ͚Ε͹ͳͥҰகͯ͠ͳ͍͔ߟ͑Δνϟϯεͩɻ
    ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔػցֶश ม਺બ୒͸೉͍͠
    

    View Slide

38. ϗʔϧυΞ΢τ๏ )PMEPVU$SPTTWBMJEBUJPO
    
    ֶशσʔλ
    ςετσʔλ
    
    GPME$SPTTWBMJEBUJPO
    ςετ
    ςετ
    ςετ
    ςετ
    ςετ
    ύλʔϯ
    ύλʔϯ
    ύλʔϯ
    ύλʔϯ
    ύλʔϯ
    o σʔλΛͭʹ෼ׂɼύλʔϯͷσʔληοτΛ༻ҙ
    o άϧʔϓ͝ͱʹֶशσʔλͰֶशɼςετσʔλͰධՁ
    o ςετσʔλͰͷධՁͷฏۉ஋ΛऔΕ͹ɼશମͷධՁ͕Մೳ
    o શσʔλΛֶशσʔλͱςετσʔλͭʹ෼ׂ
    o ֶशσʔλͰϞσϧΛ࡞੒͠ɼςετσʔλͰධՁ
    'JOFQPJOU0WFSGJUUJOHͱDSPTTWBMJEBUJPO
    ϞσϧΛֶशσʔλʹϑΟοςΟϯάͤ͞Ε͹ͤ͞Δ΄Ͳɼ
    ֎෦ͷσʔλ΁ͷ༧ଌਫ਼౓ ൚Խੑೳ
    ͕མͪͯ͠·͏ɼ
    0WFSGJUUJOHͱݺ͹ΕΔ໰୊͕ى͜Δɻ͜ΕΛආ͚Δʹ͸
    खݩͷશσʔλΛֶशσʔλͱςετσʔλʹ෼͚ɼֶशͱ
    ϞσϧͷධՁΛผͰߦ͏ඞཁ͕͋Δɻ
    *-43QΑΓҾ༻
    IUUQGBDVMUZNBSTIBMMVTDFEVHBSFUIKBNFT*4-
    

    View Slide

39. "QQFOEJY
    

    View Slide

40. ˆ =
    1
    |{i : Wi
    = 1, Xi
    L}|
    Yi
    {i:Wi=1,Xi L}
    1
    |{i : Wi
    = 0, Xi
    L}|
    Yi
    {i:Wi=0,Xi L}
    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
    JODPNF
    5SFBUNFOU $POUSPM
    1PQVMBUJPO
    $PODFQUPG$BVTBM5SFF
    3$5 $BVTBM5SFF
    ZFT
    DIJME
    OP q 1FPQMFJOBTBNFMFBGIBWF
    TJNJMBSDIBSBDUFSJTUJDT
    Ø "TJGUIFZBSFDPNFGSPN
    SBOEPNJ[FEFYQFSJNFOUT
    = E[Yi
    |Wi
    = 1] E[Yi
    |Wi
    = 0]
    AAAC1HicSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKxQsoxZQklirYKrhGR8Zn1oTHZ9oaxiroInENYuMFlA30DMBAAZNhCGUoM0BBQL7AOYYYhhSGfIZkhlKGXIZUhjyGEiA7hyGRoRgIoxkMGQwYCoBisQzVQLEiICsTLJ/KUMvABdRbClSVClSRCBTNBpLpQF40VDQPyAeZWQzWnQy0JQeIi4A6FRhUDa4arDT4bHDCYLXBS4M/OM2qBpsBckslkE6C6E0tiOfvkgj+TlBXLpAuYchA6MLr5hKGNAYLsFszgW4vAIuAfJEM0V9WNf1zsFWQarWawSKD10D3LzS4aXAY6IO8si/JSwNTg2bjcQ/MJUlACAoH3KFXDVSRC5bnAkanIXrkYTJCjfQs9QwCTZQdnKDxysEgzaDEoAGMPHMGBwYPhgCGUKAFHQzrGLYz7GAKZ6plamZqhShlYoTqEWZAAUx9AOuypWk=
    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
    "WFSBHF5SFBUNFOU&GGFDU "WFSBHF5SFBUNFOU&GGFDUBUFBDIMFBG
    :PVUDPNF
    8JGUSFBUFE
    8JGDPOUSPM
    ZFT OP
    q %PVCMFTBNQMFUSFFT
    6TFTBNQMFTUPCVJME
    USFFTPUIBUUIFFGGFDUNPTU
    EJGGFST
    6TFSFTUPGTBNQMFTUP
    FTUJNBUFUSFBUNFOUFGGFDU
    BWPJEJOHPWFSGJUUJOH
    
    

    View Slide

41. -B5F9
    &<:?\B^>&<:?\B^>aTVN@M&<:c-M ">1S<-M>aTVN@M&<:c-M ">1S<-M>
    E[Y a=1] E[Y a=0] =
    X
    l
    E[Y |L = l, A = 1]Pr[L = l]
    X
    l
    E[Y |L = l, A = 0]Pr[L = l]
    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
    Pr[Y = 1|X = 1] Pr[Y = 1|X = 0]
    AAACAHicdZDLSsNAFIYn9VbrLerChZvBIrgxJGm9dFEounFZwV4kDWUynbRDJxdmJkKJ3fgqblwo4tbHcOfbOGmrqOgPAx//OYcz5/diRoU0zXctNze/sLiUXy6srK6tb+ibW00RJRyTBo5YxNseEoTRkDQklYy0Y05Q4DHS8obnWb11Q7igUXglRzFxA9QPqU8xksrq6jt17lxXrdt21XIPv9h0u3rRNGzzuFQqwwwqduVoCpZdgZZhTlQEM9W7+lunF+EkIKHEDAnhWGYs3RRxSTEj40InESRGeIj6xFEYooAIN50cMIb7yulBP+LqhRJO3O8TKQqEGAWe6gyQHIjftcz8q+Yk0j91UxrGiSQhni7yEwZlBLM0YI9ygiUbKUCYU/VXiAeIIyxVZgUVwuel8H9o2oZVMuzLcrF2NosjD3bBHjgAFjgBNXAB6qABMBiDe/AInrQ77UF71l6mrTltNrMNfkh7/QBzZpUL
    1S<:c9>1S<:c9>
    :aOPUaaQFSQaaaaQFSQ"c-
    Y 6?
    ? A|L
    AAACB3icdZBNS8MwGMdTX+d8q3oUJHMInkrbidPb1IsHDxPci6xlpFm2haVpSVJh1N28+FW8eFDEq1/Bm9/GbKugog+E/Pj/n4fk+Qcxo1LZ9ocxMzs3v7CYW8ovr6yurZsbm3UZJQKTGo5YJJoBkoRRTmqKKkaasSAoDBhpBIOzsd+4IULSiF+pYUz8EPU47VKMlJba5s419HikvIIXExHrq5AhPIG38KJtFm3LKZds9xhO4dDJwHWgY9mTKoKsqm3z3etEOAkJV5ghKVuOHSs/RUJRzMgo7yWSxAgPUI+0NHIUEumnkz1GcE8rHdiNhD5cwYn6fSJFoZTDMNCdIVJ9+dsbi395rUR1j/yU8jhRhOPpQ92EQRXBcSiwQwXBig01ICyo/ivEfSQQVjq6vA7ha1P4P9RdyylZ7uVBsXKaxZED22AX7AMHlEEFnIMqqAEM7sADeALPxr3xaLwYr9PWGSOb2QI/ynj7BGIbl7o=
    = E[Y |A = 1] E[Y |A = 0]
    AAAB+3icdZDLSsNAFIYn9VbrLdalm8EiuDEkqVhdFKoiuKxgWyUNZTKdtEMnF2YmYol9FTcuFHHri7jzbZw2EVT0h4GP/5zDOfN7MaNCmuaHVpibX1hcKi6XVlbX1jf0zXJbRAnHpIUjFvFrDwnCaEhakkpGrmNOUOAx0vFGZ9N655ZwQaPwSo5j4gZoEFKfYiSV1dPL9XPn5v6kbrn7GZhuT6+YhlWrmvYxzODQysG2oGWYM1VArmZPf+/2I5wEJJSYISEcy4ylmyIuKWZkUuomgsQIj9CAOApDFBDhprPbJ3BXOX3oR1y9UMKZ+30iRYEQ48BTnQGSQ/G7NjX/qjmJ9I/clIZxIkmIs0V+wqCM4DQI2KecYMnGChDmVN0K8RBxhKWKq6RC+Pop/B/atmFVDfvyoNI4zeMogm2wA/aABWqgAS5AE7QABnfgATyBZ22iPWov2mvWWtDymS3wQ9rbJ9FrkwQ=
    &<:c">&<:c">
    (E[Y a=1] E[Y a=0]) (
    X
    l
    E[Y |Z = z, A = 1]Pr[Z = z]
    X
    l
    E[Y |Z = z, A = 0]Pr[Z = z])
    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
    &<:?\B^>&<:?\B^>
    aTVN@M&<:c;[ ">1S<;[>aTVN@M&<:c;[ ">1S<;[>
    
    

    View Slide

42. -B5F9
    b(X) = E[Y |A = 1, L]
    AAAB+XicdVDJSgNBEO2JW4zbqEcvjUGIIGEmm8khEBXBg4cIZpFkCD2dnqRJz0J3TyCM+RMvHhTx6p9482/sSSKo6IOCx3tVVNWzA0aFNIwPLbG0vLK6llxPbWxube/ou3tN4Ycckwb2mc/bNhKEUY80JJWMtANOkGsz0rJHF7HfGhMuqO/dyklALBcNPOpQjKSSerpuZ9rH1cvO3f1Z1TyB11ZPTxvZXL6cL1ZgTEqlgjEjlVLRhGbWmCENFqj39Pdu38ehSzyJGRKiYxqBtCLEJcWMTFPdUJAA4REakI6iHnKJsKLZ5VN4pJQ+dHyuypNwpn6fiJArxMS1VaeL5FD89mLxL68TSqdsRdQLQkk8PF/khAxKH8YxwD7lBEs2UQRhTtWtEA8RR1iqsFIqhK9P4f+kmcua+WzuppCunS/iSIIDcAgywASnoAauQB00AAZj8ACewLMWaY/ai/Y6b01oi5l98APa2ycIV5H9
    C 9
    &<:c" ->
    ⇡(L) = Pr(A = 1|L)
    AAAB+3icdVDLSgMxFM3UV62vsS7dBIvQbspMX7aLQtWNiy4q2Ae0pWTSTBuayQxJRixjf8WNC0Xc+iPu/BvTh6CiBy4czrmXe+9xAkalsqwPI7a2vrG5Fd9O7Ozu7R+Yh8mW9EOBSRP7zBcdB0nCKCdNRRUjnUAQ5DmMtJ3J5dxv3xIhqc9v1DQgfQ+NOHUpRkpLAzPZC2i6noFV2BDp86p9X88MzJSVzeXL+WIFzkmpVLAWpFIq2tDOWgukwAqNgfneG/o49AhXmCEpu7YVqH6EhKKYkVmiF0oSIDxBI9LVlCOPyH60uH0GT7UyhK4vdHEFF+r3iQh5Uk49R3d6SI3lb28u/uV1Q+WW+xHlQagIx8tFbsig8uE8CDikgmDFppogLKi+FeIxEggrHVdCh/D1KfyftHJZO5/NXRdStYtVHHFwDE5AGtjgDNTAFWiAJsDgDjyAJ/BszIxH48V4XbbGjNXMEfgB4+0TEbiSiQ==
    aQJ -
    1S "c-
    
    E[Y a=1] =
    8
    <
    :
    E[b(L)]
    E[ AY
    ⇡(L)
    ]
    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
    &<:?\B^>aMFGUa\aCFHJO\NBUSJY^
    &>aa
    aa
    &^>
    aFOE\NBUSJY^aSJHIU
    aIBU\&^<:?\B^>aMFGUa\aCFHJO\NBUSJY^
    aGSBD\^\O^aEJTQMBZTUZMFaTVN?O@\J^aIBU\C^ -@J
    aa
    aa
    aGSBD\^\O^aEJTQMBZTUZMFaTVN?O@\J^aGSBD\"@J:@J^\aIBU\aQJ^ -@J
    ^
    aFOE\NBUSJY^aSJHIU
    ˆ
    E[Y a=1] =
    8
    >
    >
    >
    >
    >
    <
    >
    >
    >
    >
    >
    :
    1
    n
    n
    X
    i=1
    ˆ
    b(Li)
    1
    n
    n
    X
    i=1
    AiYi
    ˆ
    ⇡(Li)
    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
    ˆ
    E[Y a=1]DR =
    1
    n
    n
    X
    i=1
    {ˆ
    b(Li) +
    Ai
    ˆ
    ⇡(Li)
    (Yi
    ˆ
    b(Li))}
    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
    aIBU\&^<:?\B^>@\%3^aGSBD\^\O^aEJTQMBZTUZMFaTV
    N?O@\J^a\aIBU\C^ -@J
    
    aGSBD\"@J^\aIBU\aQJ^ -@J
    ^ :@JaIBU\C^ -@J
    
    a^
    &@\%3^&<:?\B^>>& aGSBD\^\aQJ -
    ^
    aGSBD\^\aQJ? -
    ^
    C -
    C? -
    
    >
    E[ ˆ
    E[Y a=1]DR E[Y a=1]] = E[⇡(L)(
    1
    ⇡(L)
    1
    ⇡⇤(L)
    )(b(L) b⇤(L))]
    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