統計的因果推論の紹介

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1. ౷ܭతҼՌਪ࿦ͷ঺հ matsuno 2020 ೥ 12 ݄ 19 ೔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 1 / 38
2. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 2 / 38
3. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 3 / 38
4. ͸͡Ίʹ σʔλ͔ΒɺհೖʹޮՌ͕͔͋ͬͨ൱͔ɾͲΕ͘Β͍ޮՌ͕͔͋ͬͨΛ஌Γ͍ͨ ޿ࠂΛݟͤͨ͜ͱͰߪೖʹͭͳ͕͔ͬͨ൱͔ ҩྍʹ͓͚Δ౤ༀɾखज़ͷޮՌ ࣦۀڅ෇͕ۚ࠶ब৬ઌͷڅ༩ʹٴ΅͢Өڹ ૣظڭҭʹΑΔ੒੷΁ͷӨڹ ͳͲͳͲʜ ͔͠͠ɺ୯७ʹհೖͷ༗ແͱɺ݁Ռͷ૬ؔΛݟͯ΋ਖ਼͍݁͠࿦͸ಘΒΕͳ͍ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 4 / 38
5. ૬ؔͰ͸ͩΊͳྫ ͋ΔσʔλͰ͸ɺεϚϗήʔϜͷςϨϏ CM Λݟͨάϧʔϓͷํ͕ɺςϨϏ CM Λݟͯ ͍ͳ͍άϧʔϓʹൺ΂ͯεϚϗήʔϜͷར༻͕࣌ؒগͳ͍ [2] ʷ CM ΛݟΔͱར༻͕࣌ؒݮগ͢Δɻར༻࣌ؒΛ૿΍ͨ͢Ίʹ CM Λ์ૹ͠ͳ͍ ˓ CM ΛݟΔਓ͸ςϨϏࢹௌ͕࣌ؒ௕͘ɺ΋ͱ΋ͱεϚϗͷར༻ස౓͕௿͍ ૬ؔؔ܎Ͱ͸ͳ͘ɺҼՌؔ܎ΛݟΔඞཁ͕͋Δ ςϨϏࢹௌ࣌ؒ΍εϚϗͷར༻ස౓͸ɺςϨϏ CM ΛݟΔ͔൱͔ͱɺεϚϗήʔϜͷར ༻࣌ؒͷ྆ํʹӨڹ͢Δ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 5 / 38
6. ૬ؔؔ܎ɺҼՌؔ܎ɺަབྷɺٖࣅ૬ؔ ૬ؔؔ܎ X ͕େ͖͍ͱ͖ʹ Y ͕େ͖͍ɺ·ͨ͸খ͍͞ͱ͖ɺX ͱ Y ͷؒʹ૬ؔؔ܎͕͋Δͱ ͍͏ ҼՌؔ܎ X ΛมԽͤͨ͞ͱ͖ʹ Y ΋มԽ͢Δͱ͖ɺX ͱ Y ͷؒʹҼՌؔ܎͕͋Δͱ͍͏ ͜ͷͱ͖ɺX ΛݪҼɺY Λ݁ՌͱݺͿ ަབྷ X ͱ Y ͷͦΕͧΕΛ݁Ռͱ͢Δڞ௨ͷݪҼ C ͕͋Δͱ͖ɺ͜ͷߏ଄Λަབྷͱ͍͏ ͜ͷͱ͖ C ΛަབྷҼࢠͱ͍͏ ٙࣅ૬ؔ ௚઀ͷҼՌؔ܎͕ແ͍ม਺ؒʹݱΕΔ૬ؔؔ܎ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 6 / 38
7. ٙࣅ૬ؔͷྫ [3] Nicholas Cage Swimming pool drownings Number of people who drowned by falling into a pool correlates with Films Nicolas Cage appeared in Nicholas Cage Swimming pool drownings 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 0 films 2 films 4 films 6 films 80 drownings 100 drownings 120 drownings 140 drownings tylervigen.com matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 7 / 38
8. ࠓճͷ໨ඪ ౷ܭతҼՌਪ࿦ͷجૅతͳ಺༰ͱͯ͠ɺҎԼͷ߲໨Λ঺հ ҼՌޮՌΛධՁͰ͖ΔͨΊͷ৚݅ ҼՌޮՌͷఆࣜԽ ҼՌޮՌΛධՁ͢ΔͨΊͷํ๏ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 8 / 38
9. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 9 / 38
10. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 10 / 38
11. ه๏ X: հೖର৅͕΋ͭಛ௃ྔ ࿈ଓ஋·ͨ͸ 2 ஋ͱ͢Δ ଟ࣍ݩͰ΋Α͍ ҼՌਪ࿦ͷจ຺Ͱ͸ʮڞมྔʯͱݺ͹ΕΔ Y ͱ Z ͷަབྷҼࢠ Z: հೖͷ༗ແΛද͢ม਺ Z = 0 ͳΒհೖແ͠ɺZ = 1 ͳΒհೖ͋Γ Ұൠʹ͸ X ʹґଘܾͯ͠·Δ Y : ݁Ռม਺ ࿈ଓ஋·ͨ͸ 2 ஋ Ұൠʹ͸ X ͱ Z ͷ྆ํʹґଘܾͯ͠·Δ X Z Y matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 11 / 38
12. ࣮ݧݚڀ Z ͷ Y ΁ͷޮՌΛ஌Γ͚ͨΕ͹ɺX ͕ Y , Z ʹӨڹ͠ͳ͍Α͏ʹσʔλΛऩू͢Δͷ ͕࠷΋ޮՌత X Z Y ୅දతͳํ๏ ୯७ແ࡞ҝׂΓ෇͚ɿ͢΂ͯͷݸମΛɺϥϯμϜʹॲஔ܈ (Z = 1) ͱରর܈ (Z = 0) ʹׂ Γ෇͚ ཚմ๏ɿॲஔޮՌʹӨڹ͢Δ͜ͱ͕૝ఆ͞ΕΔҼࢠʢྫɿੑผͳͲʣʹ͍ͭͯ͸ɺ྆܈Ͱό ϥϯεΛͱͬͯϥϯμϜʹׂΓ෇͚ ҰରൺֱɿഎܠҼࢠ͕ྨࣅͷݸମΛରԠͤ͞ɺॲஔ܈ͱରর܈ʹׂΓ෇͚ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 12 / 38
13. ࣮ݧݚڀ͕Ͱ͖ͳ͍৔߹ ยํͷ܈ʹ݈߁΁ͷѱӨڹ͕༧૝͞ΕΔ৔߹ ੓ࡦͷޮՌΛධՁ͍ͨ͠৔߹ Ұ෦ސ٬༏۰ͷ৘ใʹΑΔѱӨڹ͕༧૝͞ΕΔ৔߹ ࣌ؒతɾֹۚతίετʹݟ߹Θͳ͍৔߹ ݸମ΁ͷॲஔͷׂΓ෇͚ͳͲΛҼՌਪ࿦޲͚ʹίϯτϩʔϧ͠ͳ͍ = ؍࡯ݚڀ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 13 / 38
14. ൓࣮Ծ૝Ϟσϧ ൓ࣄ࣮Ϟσϧɺcounterfactual model ͱ΋ ݁Ռม਺͸ຊདྷ 2 ม਺͋Δͱߟ͑Δ Y1 : ݸମ͕ॲஔ܈ Z = 1 ʹׂΓ෇͚ΒΕͨͱ͖ʹಘΒΕΔ݁Ռ Y0 : ݸମ͕ରর܈ Z = 0 ʹׂΓ෇͚ΒΕͨͱ͖ʹಘΒΕΔ݁Ռ Y1, Y0 ͸જࡏత݁Ռม਺ͱݺ͹Εɺݪཧతʹಉ࣌ʹ͸؍ଌͰ͖ͳ͍ જࡏత݁Ռม਺Λ༻͍Δͱɺ؍ଌ஋ Y ͸ҎԼͷΑ͏ʹද͞ΕΔ Y = ZY1 + (1 − Z)Y0 matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 14 / 38
15. ҼՌޮՌͷఆٛ ॲஔͷޮՌ = ॲஔΛߦͬͨ৔߹ͱߦΘͳ͔ͬͨ৔߹ͷ݁Ռͷࠩ Y1 − Y0 Y0, Y1 ͸ݸମʹରͯ͠͸ಉ࣌ʹ؍ଌͰ͖ͳ͍ͷͰɺ฼ूஂશମͰͷޮՌΛධՁର৅ͱ ͢Δ ฏۉॲஔޮՌ (Average Treatment Effect, ATE) ATE = E[Y1 − Y0] = E[Y1] − E[Y0] ATE ΛͲͷΑ͏ʹਪఆ͢Δ͔ɺ͕౷ܭతҼՌਪ࿦ͷओͳςʔϚ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 15 / 38
16. ͦͷଞͷҼՌޮՌ ॲஔ܈ʹ͓͚ΔฏۉॲஔޮՌ (Average Treatment Effect on the Treated, ATT) ATT = E[Y1 − Y0|Z = 1] ॲஔʹΑͬͯಘΒΕͨޮՌ ରর܈ʹ͓͚ΔฏۉॲஔޮՌ (Average Treatment Effect on the Untreated, ATU) ATU = E[Y1 − Y0|Z = 0] ॲஔର৅Ͱ͸ͳ͔ͬͨूஂʹɺ΋࣮͠ࢪͨ͠ͱ͖ͷޮՌ ॲஔର৅Λ޿͛Δ΂͖͔Ͳ͏͔ͷ൑அࡐྉʹͳΔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 16 / 38
17. SUTVA ৚݅ ॲஔޮՌ͕ѻ͍΍͍͢ܗͰఆٛ͞ΕΔͨΊͷ৚݅ SUTVA (stable unit treatment value assumption) ৚݅ 1 ݸମͷજࡏత݁Ռม਺͸ɺଞͷݸମ͕ड͚Δॲஔʹґଘ͠ͳ͍ 2 ॲஔ͸ 1 ௨Γʹఆ·Δ ৚݅ 1 ʹ͍ͭͯ ૬ޓׯব͕ͳ͍͜ͱ ஫ҙਂ͍࣮ݧσβΠϯ΍ɺ৚݅ͷଥ౰ੑͷධՁ͕ඞཁ ੒Γཱͨͳ͍৔߹ʹ͸ɺ૬ޓґଘੑͷϞσϧԽ΍ɺґଘؔ܎ʹ͋ΔάϧʔϓΛ 1 ݸମͱ͠ ͯѻ͏ඞཁ͕͋Δ ৚݅ 2 ʹ͍ͭͯ ॲஔͷ಺༰Λ໌֬ʹ͢Δ͜ͱ ྫɿ෩अༀͷޮՌʜ1 ೔Կճʁ1 ճ͋ͨΓԿৣʁ৯લʁ৯ޙʁ ҎԼͰ͸ SUTVA ৚݅͸੒Γཱ͍ͬͯΔ΋ͷͱ͢Δ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 17 / 38
18. ATE ͕ਪఆͰ͖ΔͨΊͷ৚݅ ڧ͍ҙຯͰͷແࢹՄೳੑ 1 ৚݅෇͖ಠཱੑ Y0, Y1 ⊥ Z | X X ͕༩͑ΒΕΕ͹ɺY0, Y1 ͱ Z ͸ಠཱ ަབྷҼࢠ͸ X Ͱਚ͘͞Ε͍ͯͯɺଞʹॏཁͳަབྷҼࢠ͸ແ͍ 2 ৚݅෇͖ਖ਼஋ੑ 0 < P(Z = 1|X) < 1 X ͕༩͑ΒΕͨͱ͖ɺඞͣॲஔΛ͢Δʗ͠ͳ͍ͱ͍ͬͨϧʔϧ͸ଘࡏ͠ͳ͍ ೚ҙͷ X ʹ͍ͭͯ੒Γཱͭ͜ͱ͕ඞཁͳɺڧ͍৚݅ ҎԼͰ͸͜ͷ৚݅΋੒Γཱ͍ͬͯΔͱ͢Δ e(X) = P(Z = 1|X) ͸܏޲είΞͱݺ͹ΕΔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 18 / 38
19. ڧ͍ҙຯͰͷແࢹՄೳੑ͕੒Γཱͭ৔߹ͷ ATE E[Y1] ͸ɺఆ͔ٛΒ E[Y1] = ∑ Y1 Y1P(Y1) = ∑ X,Z ∑ Y1 Y1P(X, Y1, Z) ৚݅෇͖ಠཱੑ͔ΒɺP(X, Y1, Z) = P(Y1, Z|X)P(X) = P(Y1|X)P(Z|X)P(X) ͱ෼ղͰ͖ͯ E[Y1] = ∑ X,Z ∑ Y1 Y1P(Y1|X)P(Z|X)P(X) = ∑ X ∑ Y1 Y1P(Y1|X)P(X) matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 19 / 38
20. ڧ͍ҙຯͰͷແࢹՄೳੑ͕੒Γཱͭ৔߹ͷ ATE Y1 ͸ Z = 1 ͷͱ͖ͷ Y ͳͷͰɺ৚݅෇͖ਖ਼஋ੑ͕੒Γཱ͍ͬͯΕ͹ E[Y1] = ∑ X ∑ Y YP(Y |Z = 1, X)P(X) = EX [E[Y |Z = 1, X]] E[Y0] ʹ͍ͭͯ΋ಉ༷ʹܭࢉ͢Δͱ ATE = E[Y1] − E[Y0] = EX [E[Y |Z = 1, X]] − EX [E[Y |Z = 0, X]] ͱͳΓɺ؍ଌͰ͖Δྔ͔ΒਪఆՄೳʹͳΔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 20 / 38
21. ܏޲είΞͷੑ࣭ ఆཧɿόϥϯγϯά ڞมྔ X ͱׂΓ෇͚ม਺ Z ͸ɺ܏޲είΞ e(X) ͷ΋ͱͰ৚݅෇͖ಠཱ X ⊥ Z | e(X) ಉ͡܏޲είΞΛ΋ͭݸମ͕΋ͭ X ͷ෼෍͸ɺॲஔ܈ͱରর܈ͱͰ౳͍͠ ূ໌ɿ ৚݅෇͖ಠཱੑͷఆٛɿ X ⊥ Z | e(X) ⇔ P(X, Z = 1|e(X)) = P(X|e(X))P(Z = 1|e(X)) ҰํͰɺ৚݅෇͖֬཰ͷੑ࣭͔Β P(X, Z = 1|e(X)) = P(X|e(X))P(Z = 1|X, e(X)) = P(X|e(X))e(X) ͳͷͰɺP(Z = 1|e(X)) = e(X) ΛࣔͤΕ͹Α͍ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 21 / 38
22. ܏޲είΞͷੑ࣭ ূ໌ʢଓ͖ʣ ɿ P(Z = 1|e(X)) = E[Z|e(X)] ظ଴஋ͷੑ࣭ E[A] = EB[E[A|B]] ΑΓɺ P(Z = 1|e(X)) = EX|e(X) [E[Z|X]|e(X)] = EX|e(X) [e(X)|e(X)] = e(X) 2 matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 22 / 38
23. ܏޲είΞͷੑ࣭ ఆཧɿ৚݅෇͖ಠཱੑ જࡏత݁Ռม਺ Y0, Y1 ͱׂΓ෇͚ม਺ Z ͸ɺ܏޲είΞ e(X) ͷ΋ͱͰ৚݅෇͖ಠཱ Y0, Y1 ⊥ Z | e(X) ূ໌ɿ ৚݅෇͖ಠཱੑͷఆٛɿ Y0 ⊥ Z | e(X) ⇔ P(Y0, Z = 1|e(X)) = P(Y0|e(X))P(Z = 1|e(X)) = P(Y0|e(X))e(X) ҰํͰɺ৚݅෇͖֬཰ͷੑ࣭͔Β P(Y0, Z = 1|e(X)) = P(Z = 1|e(X), Y0)P(Y0|e(X)) ͳͷͰɺP(Z = 1|e(X), Y0) = e(X) ΛࣔͤΕ͹Α͍ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 23 / 38
24. ܏޲είΞͷੑ࣭ ূ໌ʢଓ͖ʣ ɿ P(Z = 1|e(X), Y0) = E[Z|e(X), Y0] ظ଴஋ͷੑ࣭ E[A] = EB[E[A|B]] ΍৚݅෇͖ಠཱੑ X ⊥ Z | e(X) Λॱʹద༻͢Δͱɺ P(Z = 1|e(X)) = EX|e(X),Y0 [E[Z|X, e(X), Y0]|e(X), Y0] = EX|e(X),Y0 [E[Z|X, Y0]|e(X), Y0] = EX|e(X),Y0 [E[Z|X]|e(X), Y0] = EX|e(X),Y0 [e(X)|e(X), Y0] = e(X) Y0 Λ Y1 ͱͯ͠΋ಉ༷ʹܭࢉͰ͖Δɻ 2 matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 24 / 38
25. ܏޲είΞΛ༻͍ͨ ATE ظ଴஋ͷੑ࣭ E[A] = EB[E[A|B]] ͱ৚݅෇͖ಠཱੑ Y0, Y1 ⊥ Z | e(X) ΑΓɺ ATE = E[Y1] − E[Y0] = Ee(X) [E[Y1|e(X)]] − Ee(X) [E[Y0|e(X)]] = Ee(X) [E[Y |Z = 1, e(X)]] − Ee(X) [E[Y |Z = 0, e(X)]] X શମͰ͸ͳ͘ɺ܏޲είΞ e(X) ͷ෼෍Ͱظ଴஋ΛͱΕ͹ ATE ͷධՁ͕Մೳ ܏޲είΞ e(X) = P(Z = 1|X) ͷਪఆʹ͸ϩδεςΟ οΫճؼ΍ɺछʑͷػցֶशϞσ ϧ͕༻͍ΒΕΔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 25 / 38
26. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 26 / 38
27. ATE ਪఆͷͨΊͷख๏ ATE = EX [E[Y |Z = 1, X]] − EX [E[Y |Z = 0, X]] = Ee(X) [E[Y |Z = 1, e(X)]] − Ee(X) [E[Y |Z = 0, e(X)]] ATE ͷਪఆ஋ ˆ τ ΛಘΔ͢ΔͨΊͷख๏ɿ 1 Ϛονϯά 2 ૚Խղੳ๏ 3 ॏΈ෇͚๏ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 27 / 38
28. Ϛονϯά ॲஔ܈ Z = 1 ͱରর܈ Z = 0 ͔Βɺڞมྔ X ͕ࣅ͍ͯΔݸମΛରԠ෇͚ ରԠ෇͚͞Εͨ k ൪໨ͷϖΞͷ݁Ռม਺Λ Y (k) 1 , Y (k) 0 ͱ͢Δͱɺ ˆ τ = 1 K K ∑ k=1 (Y (k) 1 − Y (k) 0 ) ྨࣅ౓ͷج४ɿ ڞมྔͷ੒෼͝ͱʹॏΈ෇͚ͨ͠ L1 ڑ཭ɺL2 ڑ཭ ڞมྔͷڞ෼ࢄߦྻ Σ Λ༻͍ͨϚϋϥϊϏεڑ཭ (Xi − Xj )T Σ−1(Xi − Xj ) ܏޲είΞͷࠩͷઈର஋ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 28 / 38
29. Ϛονϯά ར఺ Θ͔Γ΍͘͢ɺσʔλղੳ͕༰қ ܽ఺ Ϛονϯάͷखؒ ରԠ෇͚͞Εͳ͍σʔλͷൃੜʹΑΔαϯϓϧαΠζͷ௿Լ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 29 / 38
30. ૚ผղੳ๏ X ʹ΋ͱ͖ͮݸମΛ K ݸͷ૚ʹ෼ׂ͠ɺ֤૚ͷฏۉ஋Λ౷߹ͯ͠ਪఆ݁ՌΛಘΔ ୈ k ૚ʹׂΓ౰ͯΒΕͨݸମͷൺ཰Λ q(k)ɺॲஔ܈ Z = 1 ͱରর܈ Z = 0 ͷ݁Ռม਺ ͷฏۉ஋Λ Y (k) 1 , Y (k) 0 ͱ͢Δͱ ˆ τ = K ∑ k=1 q(k)(Y (k) 1 − Y (k) 0 ) Ϛονϯά͸૚ผղੳ๏Ͱɺ֤૚ʹॲஔ܈ͱର৅܈Λ 1 ݸମؚͣͭΉಛผͳ৔߹ ڞมྔ X ͷ࣍ݩ͕ߴ͍৔߹ʹ͸ɺ܏޲είΞʹΑΔ૚Խ͕༗ޮ ܏޲είΞ͕཭Ε͍ͯͯ΋ɺແཧ΍Γಉ͡૚ʹׂΓ౰ͯΒΕͯ͠·͏Մೳੑʹ͸஫ҙ͕ ඞཁ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 30 / 38
31. ॏΈ෇͚๏ ֬཰ e(X) Ͱॲஔ܈ʹׂΓ౰ͯΒΕΔݸମͷ E[Y1] ΁ͷد༩͸ Y1/e(X) Ͱ͋Δ͜ͱΛ ߟྀ͢Δͱ ˆ τ = 1 N N ∑ i=1 Zi Yi e(Xi ) − 1 N N ∑ i=1 (1 − Zi )Yi 1 − e(Xi ) e(X), 1 − e(X) ͕খ͍͞ͱਪఆ݁Ռ͕ෆ҆ఆʹͳΔ఺ʹ͸஫ҙ͕ඞཁ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 31 / 38
32. ൃలɿೋॏʹϩόετͳਪఆ๏ ॲஔ܈ɺରর܈ͷͦΕͧΕʹׂΓ౰ͯͨ৔߹ͷ݁Ռม਺ΛϞσϧԽ (g1(X), g0(X)) ͨ͠ ৔߹ લϖʔδͷ ATE ͷਪఆ஋Λ࣍ͷΑ͏ʹมߋͯ͠ΈΔ ˆ τ = 1 N N ∑ i=1 ( Zi Yi e(Xi ) − Zi − e(Xi ) e(Xi ) × g1(Xi ) ) − 1 N N ∑ i=1 ( (1 − Zi )Yi 1 − e(Xi ) + Zi − e(Xi ) 1 − e(Xi ) × g0(Xi ) ) ͜ͷͱ͖ɺ܏޲είΞͱ݁Ռม਺ͷ͍ͣΕ͔͕ਖ਼͘͠ਪఆͰ͖͍ͯΕ͹ɺˆ τ ͷظ଴஋͸ ATE ʹ౳͘͠ͳΔ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 32 / 38
33. ൃలɿೋॏʹϩόετͳਪఆ๏ ܏޲είΞ͕ਖ਼͘͠ਪఆͰ͖͍ͯΔ৔߹ɿE[Z|X] = e(X) E[ˆ τ] = E{X} [E[ˆ τ|{X}]] E[ˆ τ|{X}] = 1 N N ∑ i=1 ( E[Zi Yi |Xi ] e(Xi ) − E[Zi |Xi ] − e(Xi ) e(Xi ) × g1(Xi ) ) − 1 N N ∑ i=1 ( E[(1 − Zi )Yi |Xi ] 1 − e(Xi ) + E[Zi |Xi ] − e(Xi ) 1 − e(Xi ) × g0(Xi ) ) = 1 N N ∑ i=1 (E[Y1,i |Xi ] − E[Y0,i |Xi ]) −→ E[ˆ τ] = E[Y1] − E[Y0] = ATE matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 33 / 38
34. ൃలɿೋॏʹϩόετͳਪఆ๏ ݁Ռม਺͕ਖ਼͘͠ਪఆͰ͖͍ͯΔ৔߹ɿE[Y1|X] = g1(X), E[Y0|X] = g0(X) E[ZY |X] = E[ZY1|X] = E[Z|X]g1(X), E[(1 − Z)Y |X] = E[(1 − Z)Y0|X] = (1 − E[Z|X])g0(X) E[ˆ τ|{X}] = 1 N N ∑ i=1 ( E[Zi |Xi ]g1(Xi ) e(Xi ) − E[Zi |Xi ] − e(Xi ) e(Xi ) × g1(Xi ) ) − 1 N N ∑ i=1 ( (1 − E[Zi |Xi ])g0(Xi ) 1 − e(Xi ) + E[Zi |Xi ] − e(Xi ) 1 − e(Xi ) × g0(Xi ) ) = 1 N N ∑ i=1 (g1(Xi ) − g0(Xi )) −→ E[ˆ τ] = E[Y1] − E[Y0] = ATE matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 34 / 38
35. ໨࣍ 1 ͸͡Ίʹ 2 ౷ܭతҼՌਪ࿦ ఆࣜԽ ฏۉॲஔޮՌਪఆͷͨΊͷख๏ 3 ·ͱΊ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 35 / 38
36. ·ͱΊ ౷ܭతҼՌਪ࿦ͷجૅతͳ಺༰ͱͯ͠ɺҎԼͷ߲໨Λ঺հͨ͠ ҼՌޮՌΛධՁͰ͖ΔͨΊͷ৚݅ ҼՌޮՌͷఆࣜԽ ҼՌޮՌΛධՁ͢ΔͨΊͷํ๏ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 36 / 38
37. ࠓճ࿩͞ͳ͔ͬͨτϐοΫ ҼՌ୳ࡧ ػցֶशͱͷؔ࿈ NeurIPS 2020 Workshop Causal Discovery & Causality-Inspired Machine Learning https://www.cmu.edu/dietrich/causality/neurips20ws/ matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 37 / 38
38. ࢀߟจݙ [1] ؠ࡚ ֶɼ౷ܭతҼՌਪ࿦ɼே૔ॻళɼ2015ɽ [2] ؠ೾σʔλαΠΤϯεץߦҕһձɼ ʰؠ೾σʔλαΠΤϯε Volɽ3ʱ ɼؠ೾ॻళɼ2016ɽ [3] Spurious Correlations, https://www.tylervigen.com/spurious-correlations. matsuno ౷ܭతҼՌਪ࿦ͷ঺հ 2020 ೥ 12 ݄ 19 ೔ 38 / 38