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

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ͦ΋ͦ΋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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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

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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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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]) <latexit sha1_base64="TbyFkzcdmM462a/bsMEfOJUDT1o=">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</latexit> 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]) <latexit sha1_base64="C+0aXIF6flpDC77qifAQHcywCg4=">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</latexit> ;Λ৚݅෇͚ͯਪఆ͠ ى͜ΔόΠΞε ਅͷҼՌޮՌ ;Λ৚݅෇͚ͨͱ͖ͷਪఆྔ ʹ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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

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

छྨͷڞมྔͷௐ੔ํ๏ Λਪఆ HGPSNVMB 9Λ৚݅෇͚ͨͱ͖ͷ:ͷฏۉ *1XFJHIUJOH 9Λ৚͚݅ͮͨͱ͖ͷ"ͷ֬཰ b(X)
= E[Y |A = 1, L] <latexit sha1_base64="U3Qr0rmhcwAq63bACQ4zjtoQR80=">AAAB+XicdVDJSgNBEO2JW4zbqEcvjUGIIGEmm8khEBXBg4cIZpFkCD2dnqRJz0J3TyCM+RMvHhTx6p9482/sSSKo6IOCx3tVVNWzA0aFNIwPLbG0vLK6llxPbWxube/ou3tN4Ycckwb2mc/bNhKEUY80JJWMtANOkGsz0rJHF7HfGhMuqO/dyklALBcNPOpQjKSSerpuZ9rH1cvO3f1Z1TyB11ZPTxvZXL6cL1ZgTEqlgjEjlVLRhGbWmCENFqj39Pdu38ehSzyJGRKiYxqBtCLEJcWMTFPdUJAA4REakI6iHnKJsKLZ5VN4pJQ+dHyuypNwpn6fiJArxMS1VaeL5FD89mLxL68TSqdsRdQLQkk8PF/khAxKH8YxwD7lBEs2UQRhTtWtEA8RR1iqsFIqhK9P4f+kmcua+WzuppCunS/iSIIDcAgywASnoAauQB00AAZj8ACewLMWaY/ai/Y6b01oi5l98APa2ycIV5H9</latexit> ⇡(L) = Pr(A = 1|L) <latexit sha1_base64="k+7orsmNzr3o7oMsfckNEUGB8Wc=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIvQbspMX7aLQtWNiy4q2Ae0pWTSTBuayQxJRixjf8WNC0Xc+iPu/BvTh6CiBy4czrmXe+9xAkalsqwPI7a2vrG5Fd9O7Ozu7R+Yh8mW9EOBSRP7zBcdB0nCKCdNRRUjnUAQ5DmMtJ3J5dxv3xIhqc9v1DQgfQ+NOHUpRkpLAzPZC2i6noFV2BDp86p9X88MzJSVzeXL+WIFzkmpVLAWpFIq2tDOWgukwAqNgfneG/o49AhXmCEpu7YVqH6EhKKYkVmiF0oSIDxBI9LVlCOPyH60uH0GT7UyhK4vdHEFF+r3iQh5Uk49R3d6SI3lb28u/uV1Q+WW+xHlQagIx8tFbsig8uE8CDikgmDFppogLKi+FeIxEggrHVdCh/D1KfyftHJZO5/NXRdStYtVHHFwDE5AGtjgDNTAFWiAJsDgDjyAJ/BszIxH48V4XbbGjNXMEfgB4+0TEbiSiQ==</latexit> Λਪఆ C 9 ͱК 9 ͸ڞʹઢܗճؼ 0-4 ΍ϩδεςΟοΫճؼͰਪఆͰ͖Δɻ ਪఆͷࡍʹ͸ަ߲ࠩ΍DVCJDTQMJOFͳͲͰ͖Δ͚ͩଟ͘ͷ৘ใΛ੝ΓࠐΜͰɼਅͷϞσϧʹ͚͍ۙͮͨɻ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

ҼՌਪ࿦ͷަབྷௐ੔ʹ͓͍ͯػցֶशΛ༻͍Δ͜ͱͷ໰୊఺ͭ બ୒͞Εͨม਺͕ަབྷΛऔΓআ͍͍ͯΔͱ͸ݶΒͳ͍ o 9OPODPOGPVOEFS DPOGPVOEFSͳͷͰɼແବͳม਺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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

%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) <latexit sha1_base64="SAWyeqb7SU/VZlTFhtBY2turq5Q=">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</latexit> E[Y a=1] = 8 < : E[b(L)] E[ AY ⇡(L) ] <latexit sha1_base64="VWFQcEPfo3q+/Ioh3ysMoW8nmuo=">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</latexit> if IP weighting, where ⇡(L) = Pr(A = 1|L) <latexit sha1_base64="7UznOsdzYNs6BV1Kku0N3p8dan8=">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</latexit> if standardization, where b(L) = E[Y |A = 1, L] <latexit sha1_base64="wynJGMyL5y38L6O1qnGPxmfu1v4=">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</latexit> ˆ E[Y a=1]DR = 1 n n X i=1 {ˆ b(Li) + Ai ˆ ⇡(Li) (Yi ˆ b(Li))} <latexit sha1_base64="4K/yOdand7aX54fOKatdtnQbfwk=">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</latexit> ! ! ͸JOEJWJEVBM͝ͱͷ:ͷ༧ଌ஋ $ π ! ͸JOEJWJEVBM͝ͱͷ"ͷ༧ଌ஋ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

%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))} <latexit sha1_base64="4K/yOdand7aX54fOKatdtnQbfwk=">AAACY3icdVHfa9QwHE+rznlTV7e9iRA8hBvi0d5t5/YwmE7BBx+mePvBpStpLt2FpWlJ0kEX8k/65psv/h9L2xOm6BcCHz4/+CafpCVnSofhD8+/d//BysPVR721x0+ergfPNk5UUUlCp6TghTxLsaKcCTrVTHN6VkqK85TT0/TqqNFPr6lUrBDfdF3SOMeXgmWMYO2oJLhBC6zNRzs7vzD4ILJxYj58tQcok5iYyBph0ZypkuNa6ZpTpKr8QiSGOStEpg2ndvA5YduvYRd6lzDbCahknWQH5wl7c9e8jWwvCfrhcDTeG+/uwwZMJjthC/YnuxGMhmE7fbCc4yT4juYFqXIqNOFYqVkUljo2WGpGOLU9VClaYnKFL+nMQYFzqmLTdmThK8fMYVZId4SGLXs3YXCuVJ2nzpljvVB/aw35L21W6WwvNkyUlaaCdIuyikNdwKZwOGeSEs1rBzCRzN0VkgV2TWn3LU0Jv18K/w9ORsNoPBx92ekfvl/WsQqeg5dgACLwFhyCT+AYTAEBP70Vb90LvF/+mr/hb3VW31tmNsEf47+4BezEuEI=</latexit> E[ ˆ E[Y a=1]DR E[Y a=1]] = E[⇡(L)( 1 ⇡(L) 1 ⇡⇤(L) )(b(L) b⇤(L))] <latexit sha1_base64="oSC5nlSimXzUbxGuR5UWLpvA+AQ=">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</latexit> К - К - JGUSFBUNFOUNPEFM JTDPSSFDU C - C - JGPVUDPNFNPEFMJTDPSSFDU ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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))] <latexit sha1_base64="oSC5nlSimXzUbxGuR5UWLpvA+AQ=">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</latexit> 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 <latexit sha1_base64="hv9476wbATu1cwQ27cl/J6mBIy8=">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</latexit> ͭͷςΫχοΫɼTQMJUUJOHͱDSPTTGJUUJOHΛ%3FTUJNBUPSʹಋೖ͠Α͏ ࢲͷٙ໰ ʮ෼ࢄ͕খ͘͞ͳΔ࿩ͱ Ͳ͏ܨ͕Δͷ ʯ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

4UVEZ QPQVMBUJPO %3FTUJNBUPSͱػցֶशͷ૊Έ߹ΘͤʹඞཁͳςΫχοΫTQMJUUJOH &TUJNBUJPO TBNQMF 5SBJOJOH TBNQMF ! ()
& () ! () & π() ATEDR = ˆ E[Y a=1]DR ˆ E[Y a=0]DR <latexit sha1_base64="G87gY1vesnnWlBtmMRonhHp+pEs=">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</latexit> 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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

%3FTUJNBUPSͱػցֶशͷ૊Έ߹ΘͤʹඞཁͳςΫχοΫDSPTTGJUUJOH 4UVEZ QPQVMBUJPO &TUJNBUJPO TBNQMF 5SBJOJOH TBNQMF " ()
' () " () ' π() ATEDR = ˆ E[Y a=1]DR ˆ E[Y a=0]DR <latexit sha1_base64="G87gY1vesnnWlBtmMRonhHp+pEs=">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</latexit> &TUJNBUJPO TBNQMF 5SBJOJOH TBNQMF " () ' () " () ' π() ATEDR = ˆ E[Y a=1]DR ˆ E[Y a=0]DR <latexit sha1_base64="GqFQPFhe/dgoP2+vzGW16aN37kA=">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</latexit> 4QMJUUJOHͷσϝϦοτ ҼՌޮՌͷਪఆͷࡍʹ൒෼ͷσʔλΛࣺͯΔ͜ͱʹͳΓɼ΋͍ͬͨͳ͍ɻ શαϯϓϧ࢖ͬͨਪఆ஋ͷ৴པ۠ؒΑΓ޿͘ͳͬͨΓͦ͠͏ɻ $SPTTGJUUJOHͷΞΠσΞ 4QMJUUJOHΛճߦ͏͜ͱͰ5SBJOJOHͱ&TUJNBUJPOͷ໾ׂΛεΠον͠ɼ ͭͷฏۉҼՌޮՌΛग़ͦ͏ ࠷ऴతͳશαϯϓϧͰͷฏۉҼՌޮՌ͸ͭͷҼՌޮՌͷฏۉΛऔͬͯಘΔɻ 4QMJU 4QMJU Entire ATEDR = <latexit sha1_base64="mmv7qG+MQWT7XfHIdQqA3AApIks=">AAACAHicdVDLSsNAFJ3UV62vqAsXbgaL4CokragboT4KLqu0tdCGMJlO2qGTSZiZiCVk46+4caGIWz/DnX/j9CH4PDBwOOde7pzjx4xKZdvvRm5mdm5+Ib9YWFpeWV0z1zeaMkoEJg0csUi0fCQJo5w0FFWMtGJBUOgzcu0Pzkb+9Q0Rkka8roYxcUPU4zSgGCkteeZWJ/Sj27TKFRUEntSrmZeeX2XHnlm0rfKBrQF/E8eyxyiCKWqe+dbpRjgJCVeYISnbjh0rN0VCUcxIVugkksQID1CPtDXlKCTSTccBMrirlS4MIqEfV3Csft1IUSjlMPT1ZIhUX/70RuJfXjtRwZGbUh4ninA8ORQkDKoIjtqAXZ0aKzbUBGFB9V8h7iOBsNKdFXQJn0nh/6RZspyyVbrcL1ZOp3XkwTbYAXvAAYegAi5ADTQABhm4B4/gybgzHoxn42UymjOmO5vgG4zXD2P3lkU=</latexit> ATEDR from split 2 <latexit sha1_base64="wvJJrw1p+zEz+GOjB2CmxVdFPFc=">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</latexit> ATEDR from split 1 <latexit sha1_base64="JnZeEYemO1yguWzFG/DfCfOgX+c=">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</latexit> + <latexit sha1_base64="WTtpBEiRBJwtEfh8X8h0HNX3l1w=">AAAB6HicdVDJSgNBEK2JW4xb1KOXxiAIwjCTBOIx6MVjAiYGkiH0dGqSNj0L3T1CGPIFXjwo4tVP8ubf2FkE1wcFj/eqqKrnJ4Ir7TjvVm5ldW19I79Z2Nre2d0r7h+0VZxKhi0Wi1h2fKpQ8AhbmmuBnUQiDX2BN/74cubf3KFUPI6u9SRBL6TDiAecUW2k5lm/WHLsSrXmuBXym7i2M0cJlmj0i2+9QczSECPNBFWq6zqJ9jIqNWcCp4VeqjChbEyH2DU0oiEqL5sfOiUnRhmQIJamIk3m6teJjIZKTULfdIZUj9RPbyb+5XVTHZx7GY+SVGPEFouCVBAdk9nXZMAlMi0mhlAmubmVsBGVlGmTTcGE8Pkp+Z+0y7ZbscvNaql+sYwjD0dwDKfgQg3qcAUNaAEDhHt4hCfr1nqwnq2XRWvOWs4cwjdYrx/BEozp</latexit> 2 <latexit sha1_base64="/ZKRoIbcmtsrh+VCuV3vkjWA2Ug=">AAAB6HicdVDJSgNBEK2JW4xb1KOXxiB4GmaSQDwGvXhMwCyQDKGnU5O06Vno7hHCkC/w4kERr36SN//GziK4Pih4vFdFVT0/EVxpx3m3cmvrG5tb+e3Czu7e/kHx8Kit4lQybLFYxLLrU4WCR9jSXAvsJhJp6Avs+JOrud+5Q6l4HN3oaYJeSEcRDzij2kjN8qBYcuxKtea4FfKbuLazQAlWaAyKb/1hzNIQI80EVarnOon2Mio1ZwJnhX6qMKFsQkfYMzSiISovWxw6I2dGGZIglqYiTRbq14mMhkpNQ990hlSP1U9vLv7l9VIdXHgZj5JUY8SWi4JUEB2T+ddkyCUyLaaGUCa5uZWwMZWUaZNNwYTw+Sn5n7TLtluxy81qqX65iiMPJ3AK5+BCDepwDQ1oAQOEe3iEJ+vWerCerZdla85azRzDN1ivH8uujPA=</latexit> ৴པ۠ؒ͸#PPUTUSBQͰܭࢉ͢Δ GPME DSPTTWBMJEBUJPOͱ ಉ͡ΞΠσΞ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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 %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

෼ࢄരൃͷ໰୊ o ཧ࿦తʹ͸ʮ%3ਪఆྔͷ෼ࢄʯʮC 9 ΍К 9 ͕ط஌ͷ৔߹ʹಘΒΕΔਪఆྔͷ෼ࢄʯͱͳΔ͸ͣɻ o ͔͠͠ɼ෼ࢄരൃ͕ى͍ͬͯ͜ΔՄೳੑ΋͋Γɼਪఆ஋ͷ৴པ͕۠ؒҙຯͷͳ͍΋ͷʹͳ͍ͬͯΔՄೳੑ͕͋Δɻ
࠷ڧͬΆ͍%3FTUJNBUPS ػցֶशʹ΋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 %3FTUJNBUPS ػցֶश͸Ϛγϯύϫʔతʹݫ͍͠ o UJNFWBSZJOHUSFBUNFOUT͕͋Δ෼ੳɼಛʹTVSWJWBMBOBMZTJTͳͲΛѻ͏ͷ͸ݱ࣮తͰ͸ͳ͍ o ͜ͷล͸1BSUͰ΍Δͦ͏Ͱ͢ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

෼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) <latexit sha1_base64="cHBeGYgCtymPY2wBhTlT3CFnxrE=">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</latexit> ʮ͜Μͳ݁Ռ͓͔͍͠ʂ෼ࢄരൃΛى͜͢ม਺Λআ͍ͯ෼ੳΛ΍Γ௚ͧ͢ʂʯͱࢥͬͯ͠·͏͕ɼ ෼ੳલʹม਺Λዞҙతʹ࡟আ͢Ε͹ɼ࠷ऴతʹWBMJEͳ৴པ۠ؒΛಘΒΕΔอূ͕ͳ͘ͳΔɻ Ұମզʑ͸Ͳ͏͢Ε͹͍͍ͷ͔ʁ ҼՌਪ࿦ͱ༧ଌ όΠΞεΛ΋ͨΒ͢ڞมྔ ަབྷௐ੔ʹػցֶश %3ਪఆྔ ػցֶश ม਺બ୒͸೉͍͠

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

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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} <latexit sha1_base64="fbj86PMEYO9Feux7/kb4wfsiyik=">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</latexit> <latexit sha1_base64="fbj86PMEYO9Feux7/kb4wfsiyik=">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</latexit> <latexit sha1_base64="fbj86PMEYO9Feux7/kb4wfsiyik=">AAADTHicjVHLThRBFL3dKOD4YMSNCZuOE4gLHW8bEoGEhOiGBQsejjOGhk51UQMV+pXu6kmg6B/wB0x0pYkL4me4caUrF3yCcYnGhZp4u6eN4alV6apb555z+1RdL/ZlqhAPDHPgwsXBoeFLtctXrl4bqV8ffZJGWcJFi0d+lHQ8lgpfhqKlpPJFJ04ECzxftL3tR0W+3RNJKqPwsdqJxVrANkPZlZwpgtz6C2eLKe0oluXWrOV0E8a1nes9R1typu3KWfuO1XGlI0Nrwcn3cifNgnX91JW5q0/n5Nbd0+rgf9Q5wsndegObWA7rZGBXQQOqsRjVP4ADGxABhwwCEBCCotgHBinNVbABISZsDTRhCUWyzAvIoUbajFiCGIzQbVo36bRaoSGdi5ppqeb0F5++hJQWjOMn3MdDfI9v8TP+PLOWLmsUXnZo9/paEbsjz26ufP+nKqBdwdZf1bmeFXRhqvQqyXtcIsUteF/f231+uDKzPK4n8DV+If+v8ADf0Q3C3lf+ZkksvzzHzx8nHs3iHc5+PU2MoMzXqJ328eadDFr3m9NNXJpszD2s+joMY3ALblPzHsAczMMitIAbYEwY9ww0P5rfzB/mrz7VNCrNDTgyBgZ/AzZd0Kw=</latexit> 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] <latexit sha1_base64="NwrBkY1TLO5+b8+S66lmZE8QZhc=">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</latexit> <latexit sha1_base64="NwrBkY1TLO5+b8+S66lmZE8QZhc=">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</latexit> <latexit sha1_base64="NwrBkY1TLO5+b8+S66lmZE8QZhc=">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</latexit> "WFSBHF5SFBUNFOU&GGFDU "WFSBHF5SFBUNFOU&GGFDUBUFBDIMFBG :PVUDPNF 8JGUSFBUFE 8JGDPOUSPM ZFT OP q %PVCMFTBNQMFUSFFT 6TFTBNQMFTUPCVJME USFFTPUIBUUIFFGGFDUNPTU EJGGFST 6TFSFTUPGTBNQMFTUP FTUJNBUFUSFBUNFOUFGGFDU BWPJEJOHPWFSGJUUJOH

-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] <latexit sha1_base64="pRu71KnGzzc4c4mQn982wRBYzmk=">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</latexit> Pr[Y = 1|X = 1] Pr[Y = 1|X = 0] <latexit sha1_base64="bhYQ66KiEoipirfvvJ/nnj864SY=">AAACAHicdZDLSsNAFIYn9VbrLerChZvBIrgxJGm9dFEounFZwV4kDWUynbRDJxdmJkKJ3fgqblwo4tbHcOfbOGmrqOgPAx//OYcz5/diRoU0zXctNze/sLiUXy6srK6tb+ibW00RJRyTBo5YxNseEoTRkDQklYy0Y05Q4DHS8obnWb11Q7igUXglRzFxA9QPqU8xksrq6jt17lxXrdt21XIPv9h0u3rRNGzzuFQqwwwqduVoCpZdgZZhTlQEM9W7+lunF+EkIKHEDAnhWGYs3RRxSTEj40InESRGeIj6xFEYooAIN50cMIb7yulBP+LqhRJO3O8TKQqEGAWe6gyQHIjftcz8q+Yk0j91UxrGiSQhni7yEwZlBLM0YI9ygiUbKUCYU/VXiAeIIyxVZgUVwuel8H9o2oZVMuzLcrF2NosjD3bBHjgAFjgBNXAB6qABMBiDe/AInrQ77UF71l6mrTltNrMNfkh7/QBzZpUL</latexit> 1S<:c9>1S<:c9> :aOPUaaQFSQaaaaQFSQ"c- Y 6? ? A|L <latexit sha1_base64="5gZ0lUjkQL+KStm/Myd7TZ5nyHA=">AAACB3icdZBNS8MwGMdTX+d8q3oUJHMInkrbidPb1IsHDxPci6xlpFm2haVpSVJh1N28+FW8eFDEq1/Bm9/GbKugog+E/Pj/n4fk+Qcxo1LZ9ocxMzs3v7CYW8ovr6yurZsbm3UZJQKTGo5YJJoBkoRRTmqKKkaasSAoDBhpBIOzsd+4IULSiF+pYUz8EPU47VKMlJba5s419HikvIIXExHrq5AhPIG38KJtFm3LKZds9xhO4dDJwHWgY9mTKoKsqm3z3etEOAkJV5ghKVuOHSs/RUJRzMgo7yWSxAgPUI+0NHIUEumnkz1GcE8rHdiNhD5cwYn6fSJFoZTDMNCdIVJ9+dsbi395rUR1j/yU8jhRhOPpQ92EQRXBcSiwQwXBig01ICyo/ivEfSQQVjq6vA7ha1P4P9RdyylZ7uVBsXKaxZED22AX7AMHlEEFnIMqqAEM7sADeALPxr3xaLwYr9PWGSOb2QI/ynj7BGIbl7o=</latexit> = E[Y |A = 1] E[Y |A = 0] <latexit sha1_base64="Z//X5VGyd3QDrmVyYHN1D2KOKMg=">AAAB+3icdZDLSsNAFIYn9VbrLdalm8EiuDEkqVhdFKoiuKxgWyUNZTKdtEMnF2YmYol9FTcuFHHri7jzbZw2EVT0h4GP/5zDOfN7MaNCmuaHVpibX1hcKi6XVlbX1jf0zXJbRAnHpIUjFvFrDwnCaEhakkpGrmNOUOAx0vFGZ9N655ZwQaPwSo5j4gZoEFKfYiSV1dPL9XPn5v6kbrn7GZhuT6+YhlWrmvYxzODQysG2oGWYM1VArmZPf+/2I5wEJJSYISEcy4ylmyIuKWZkUuomgsQIj9CAOApDFBDhprPbJ3BXOX3oR1y9UMKZ+30iRYEQ48BTnQGSQ/G7NjX/qjmJ9I/clIZxIkmIs0V+wqCM4DQI2KecYMnGChDmVN0K8RBxhKWKq6RC+Pop/B/atmFVDfvyoNI4zeMogm2wA/aABWqgAS5AE7QABnfgATyBZ22iPWov2mvWWtDymS3wQ9rbJ9FrkwQ=</latexit> &<: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]) <latexit sha1_base64="C+0aXIF6flpDC77qifAQHcywCg4=">AAACPnicdZBLSwMxFIUzvq2vqks3wSK0YMtkFB+LQlUElxXsQ6djyaSpBjMPkoxQx/4yN/4Gdy7duFDErUvTdgo+LwQ+zrmXm3vckDOpTPPRGBkdG5+YnJpOzczOzS+kF5eqMogEoRUS8EDUXSwpZz6tKKY4rYeCYs/ltOZeHfT82jUVkgX+ieqE1PHwhc/ajGClpWa6kj20T89jXERdJ5+g2XVyMA+zDRl5TQ61entWvFmHe0XklIWt2cn/9syhl5tLNdMZs4C2N0xrFw5gCyVgIYgKZr8yIKlyM/3QaAUk8qivCMdS2sgMlRNjoRjhtJtqRJKGmFzhC2pr9LFHpRP3z+/CNa20YDsQ+vkK9tWvEzH2pOx4ru70sLqUP72e+JdnR6q948TMDyNFfTJY1I44VAHsZQlbTFCieEcDJoLpv0JyiQUmSifeC2F4KfwfqlYBbRSs481MaT+JYwqsgFWQBQhsgxI4AmVQAQTcgSfwAl6Ne+PZeDPeB60jRjKzDL6V8fEJy+yqtQ==</latexit> &<:?\B^>&<:?\B^> aTVN@M&<:c;[ ">1S<;[>aTVN@M&<:c;[ ">1S<;[>

-B5F9 b(X) = E[Y |A = 1, L] <latexit sha1_base64="U3Qr0rmhcwAq63bACQ4zjtoQR80=">AAAB+XicdVDJSgNBEO2JW4zbqEcvjUGIIGEmm8khEBXBg4cIZpFkCD2dnqRJz0J3TyCM+RMvHhTx6p9482/sSSKo6IOCx3tVVNWzA0aFNIwPLbG0vLK6llxPbWxube/ou3tN4Ycckwb2mc/bNhKEUY80JJWMtANOkGsz0rJHF7HfGhMuqO/dyklALBcNPOpQjKSSerpuZ9rH1cvO3f1Z1TyB11ZPTxvZXL6cL1ZgTEqlgjEjlVLRhGbWmCENFqj39Pdu38ehSzyJGRKiYxqBtCLEJcWMTFPdUJAA4REakI6iHnKJsKLZ5VN4pJQ+dHyuypNwpn6fiJArxMS1VaeL5FD89mLxL68TSqdsRdQLQkk8PF/khAxKH8YxwD7lBEs2UQRhTtWtEA8RR1iqsFIqhK9P4f+kmcua+WzuppCunS/iSIIDcAgywASnoAauQB00AAZj8ACewLMWaY/ai/Y6b01oi5l98APa2ycIV5H9</latexit>
C 9 &<:c" -> ⇡(L) = Pr(A = 1|L) <latexit sha1_base64="k+7orsmNzr3o7oMsfckNEUGB8Wc=">AAAB+3icdVDLSgMxFM3UV62vsS7dBIvQbspMX7aLQtWNiy4q2Ae0pWTSTBuayQxJRixjf8WNC0Xc+iPu/BvTh6CiBy4czrmXe+9xAkalsqwPI7a2vrG5Fd9O7Ozu7R+Yh8mW9EOBSRP7zBcdB0nCKCdNRRUjnUAQ5DmMtJ3J5dxv3xIhqc9v1DQgfQ+NOHUpRkpLAzPZC2i6noFV2BDp86p9X88MzJSVzeXL+WIFzkmpVLAWpFIq2tDOWgukwAqNgfneG/o49AhXmCEpu7YVqH6EhKKYkVmiF0oSIDxBI9LVlCOPyH60uH0GT7UyhK4vdHEFF+r3iQh5Uk49R3d6SI3lb28u/uV1Q+WW+xHlQagIx8tFbsig8uE8CDikgmDFppogLKi+FeIxEggrHVdCh/D1KfyftHJZO5/NXRdStYtVHHFwDE5AGtjgDNTAFWiAJsDgDjyAJ/BszIxH48V4XbbGjNXMEfgB4+0TEbiSiQ==</latexit> aQJ - 1S "c- E[Y a=1] = 8 < : E[b(L)] E[ AY ⇡(L) ] <latexit sha1_base64="VWFQcEPfo3q+/Ioh3ysMoW8nmuo=">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</latexit> &<:?\B^>aMFGUa\aCFHJO\NBUSJY^ &<C - >aa aa &<aGSBD\":^\aQJ - ^> aFOE\NBUSJYâSJHIU aIBU\&^<:?\B^>aMFGUa\aCFHJO\NBUSJY^ aGSBD\^\OâEJTQMBZTUZMFaTVN?O@\JâIBU\C^ -@J aa aa aGSBD\^\OâEJTQMBZTUZMFaTVN?O@\JâGSBD\"@J:@J^\aIBU\aQJ^ -@J ^ aFOE\NBUSJYâSJHIU ˆ E[Y a=1] = 8 > > > > > < > > > > > : 1 n n X i=1 ˆ b(Li) 1 n n X i=1 AiYi ˆ ⇡(Li) <latexit sha1_base64="SAWyeqb7SU/VZlTFhtBY2turq5Q=">AAACnnicjVHbatwwEJXdW7q9bdrHvIguhfTF2LvJJnkIJCktpReaQjfZsHKMLMu7Q2TZSHKJEf6s/kjf+jfVejfQlhY6IDicmTMzOpNWArQJwx+ef+v2nbv3Nu73Hjx89PhJf/PpmS5rxfiElaJU05RqLkDyiQEj+LRSnBap4Ofp1atl/vwrVxpK+cU0FY8LOpeQA6PGUUn/G1lQY1+3s4tLSw+jNj7ERPDcEEtSPgdpC2oUXLc9kivKbNRa2ZIMdCVoo00jONF1cSkTC07c9Urb7Q8JvMSE4B4h/6frao4TuEigtV0XUsGqj5vMZXazBlEwX5gg6Q/CYDjaH+0e4CUYj3fCDhyMdyMcBWEXA7SO06T/nWQlqwsuDRNU61kUVia2VBlggrsZteYVZVd0zmcOSlpwHdvO3ha/cEyG81K5Jw3u2F8VlhZaN0XqKt2aC/1nbkn+LTerTb4fW5BVbbhkq0F5LbAp8fJWOAPFmRGNA5QpcLtitqDOK+Mu2nMm3PwU/xucDYNoFAw/7wyOTtZ2bKAt9BxtowjtoSP0Fp2iCWLelnfsvfPe+9h/43/0P61KfW+teYZ+C3/6E7sM0Os=</latexit> ˆ E[Y a=1]DR = 1 n n X i=1 {ˆ b(Li) + Ai ˆ ⇡(Li) (Yi ˆ b(Li))} <latexit sha1_base64="4K/yOdand7aX54fOKatdtnQbfwk=">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</latexit> aIBU\&^<:?\B^>@\%3âGSBD\^\OâEJTQMBZTUZMFaTV N?O@\Jâ\aIBU\C^ -@J aGSBD\"@J^\aIBU\aQJ^ -@J ^ :@JaIBU\C^ -@J a^ &<aIBU\&^<:?\B^>@\%3^&<:?\B^>>&<aQJ - 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))] <latexit sha1_base64="oSC5nlSimXzUbxGuR5UWLpvA+AQ=">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</latexit>

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

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

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