疫学のための確率の基礎

Causal Inference: What If』勉強会の第0回目の資料です．
疫学の理論を読み解くための確率の基礎をまとめました．

Transcript

1. Shuntaro Sato
   (Sato@ੜ෺౷ܭՈ)
   ӸֶͷͨΊͷ֬཰ͷجૅ
   Causal Inference: What Ifษڧձ
   

   View Slide

2. ࣭໰͸ʁ
   2
   • Slack: ษڧձதͷ࣭໰ശνϟϯωϧʹ౤ߘ͍ͯͩ͘͠͞
   • ܗࣜ͸ͳ͍Ͱ͢
   • ษڧձதҎ֎ͷ࣭໰͸ɼSlack: ΈΜͳ΁ͷ࣭໰νϟϯωϧʹ౤ߘ͍ͯͩ͘͠͞
   

   View Slide

3. Roadmap (1)
   3
   Population of interest
   vs. vs.
   Causation Association
   Treated Untreated
   E[Ya=1] E[Ya=0] E[Y|A = 1] E[Y|A = 0]
   E[Ya=1] − E[Ya=0] E[Y|A = 1] − E[Y|A = 0]
   Average causal effect Association measure
   Goal
   Hernán MA, Robins JM (2020). Causal Inference: What If. Boca Raton: Chapman & Hall/CRC. Figure 1.1
   

   View Slide

4. Roadmap (2)
   4
   Average causal effect ͱ Association measureͷ ߏ੒ཁૉΛཧղ͢Δ
   Goal
   Average causal effect
   Association measure
   E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
   E[Y|A = 1] − E[Y|A = 0]
   ظ଴஋ ظ଴஋ͷઢܗੑ
   ฏۉ ֬཰
   ֬཰ม਺
   ৚͖݅ͭظ଴஋
   ৚͖݅ͭ֬཰
   Marginal
   Conditional
   w ಉ࣌֬཰
   w ಠཱੑ
   w $IBJOSVMF
   w શ֬཰ͷެࣜ
   पลԽ
   ճؼ
   पลԽ
   

   View Slide

5. ஫ҙ
   5
   • ࠓճͷษڧձͷ໨త͸ʮ֬཰ͷجૅʯͷཧղ
   •γϯϓϧͳه๏Λ༻͍Δ
   • જࡏΞ΢τΧϜ౳ͷCausal Inferenceಛ༗ͷه๏͸Ͱ͖Δ͚ͩ༻͍ͳ͍
   E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
   E[X] − E[Y] = E[X − Y]
   E[Y1
   ] − E[Y2
   ] = E[Y1
   − Y2
   ]
   or
   

   View Slide

6. ·ͱΊʢ͜Ε͚ͩ͸֮͑ͯʂʣ
   6
   1. ڵຯ͋Δม਺ͷظ଴஋ = ڵຯ͋Δม਺ͷฏۉ
   ɹೋ஋σʔλͷ৔߹ɼظ଴஋ = ฏۉ = ׂ߹
   2. ʰ࿨ͷظ଴஋͸ɼظ଴஋ͷ࿨ʱʢ਺ֶΨʔϧ ཚ୒ΞϧΰϦζϜΑΓʣ
   3. ৚͖݅ͭظ଴஋͸ɼαϒάϧʔϓ಺Ͱͷظ଴஋
   E[X + Y] = E[X] + E[Y]
   E[Y|A = 1]
   E[Y|A = 0]
   Y A
   1
   1
   1
   0
   0
   0
   

   View Slide

7. େࣄͳ༻ޠ
   7
   ೔ຊޠ ӳޠ ه๏ͷྫ
   ࣄ৅ Event
   ഉ൓ Disjoint
   ཭ࢄܕ Discrete type
   ࿈ଓܕ Continuous type
   ֬཰ Probability
   ֬཰ม਺ Random variable
   ֬཰෼෍ Probability distribution
   ֬཰ີ౓ؔ਺ Probability density function:
   PDF
   ظ଴஋ Expected value
   ظ଴஋ͷઢܗੑ Linearity of expectation
   ৚͖݅ͭ֬཰ Conditional probability
   ಉ࣌֬཰ Joint probability
   ඪ४Խ Standardization
   ಠཱੑ Independent
   ৚͖݅ͭಠཱ Conditional independent
   શ֬཰ͷެࣜ Law of total probability
   पล֬཰ Marginal probability
   ࿈࠯ެࣜ Chain rule
   A
   Pr(X), Pr(X = x)
   X
   Pr
   f
   E[X]
   Pr[Y |A = a]
   Pr[Y |A], Pr(Y = y, A = a)
   Y⊥
   ⊥ A|L
   Y⊥
   ⊥ A
   

   View Slide

8. ظ଴஋Λ෼ղ͢Δʢ1ʣ
   8
   Average causal effect
   Association measure
   E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
   E[Y|A = 1] − E[Y|A = 0]
   ظ଴஋ ظ଴஋ͷઢܗੑ
   ฏۉ ֬཰
   ֬཰ม਺
   ৚͖݅ͭظ଴஋
   ৚͖݅ͭ֬཰
   Marginal
   Conditional
   w ಉ࣌֬཰
   w ಠཱੑ
   w $IBJOSVMF
   w શ֬཰ͷެࣜ
   पลԽ
   ճؼ
   पลԽ
   

   View Slide

9. ظ଴஋Λ෼ղ͢Δʢ2ʣ
   9
   E[X] =
   ∞
   ∑
   k=1
   xk
   Pr(X = xk
   )
   ظ଴஋ͷఆٛ
   ֬཰ม਺
   ؍ଌσʔλ
   ֬཰
   ֬཰ม਺ɹ ͷظ଴஋ɹɹ Λ࣍ࣜͰఆٛ͢Δɽ͜͜Ͱɼ
   • ɹ͸ɼ֬཰ม਺ ɹ ͕ͱΔ஋
   • ɹɹɹɹ ͸ɼ֬཰ม਺ɹ ͕ɹ ʹ౳͘͠ͳΔ֬཰Λද͢
   xk
   X
   X E[X]
   Pr(X = xk
   ) X xk
   

   View Slide

10. 3௨Γͷ֬཰ͷҙຯ
    10
    ݹయత֬཰ ౷ܭత֬཰ ެཧత֬཰
    ಉఔ౓ʹ͔֬Β͍͠
    ͢΂ͯͷ৔߹ͷ਺ʹର͢
    Δɼ͋Δࣄ৅ͷى͜Δ৔
    ߹ͷ਺ͷൺΛ֬཰ͱ͢Δ
    શମͷ਺ʹର͢Δɼ
    ͋Δࣄ৅ͷى͜Δ਺ͷൺ
    Λ֬཰ͱ͢Δ
    ֬཰ͷެཧʹΑͬͯఆΊ
    ͨ֬཰
    ݱ୅਺ֶͰͷ֬཰ͷఆٛ
    ݹయత֬཰
    ެཧత֬཰
    

    View Slide

11. ֬཰ͷެཧ
    11
    ɹΛू߹ͱ͠ɼɹɼɹΛɹͷ෦෼ू߹ͱ͢Δɽ
    ɹ Λɹ ͷ෦෼ू߹͔Β࣮਺΁ͷؔ਺ͱ͢Δɽ
    ؔ਺ɹ ͕ҎԼͷ3ͭͷެཧΛຬͨ͢ͱ͠Α͏ɽ
    • ू߹ɹ Λඪຊۭؒͱݺͼɼ
    • ɹ ͷ෦෼ू߹Λࣄ৅ͱݺͼɼ
    • ؔ਺ɹɹΛ֬཰෼෍ͱݺͼɼ
    • ࣮਺ɹɹɹΛɹ ͕ى͖Δ֬཰ͱݺͿ
    Ω
    Ω
    A B Ω
    Pr
    Pr
    0 ≤ Pr(A) ≤ 1
    Pr(Ω) = 1
    A ∩ B = ϕ Pr(A ∪ B) = Pr(A) + Pr(B)
    ͳΒ͹ɼ
    ͜ͷͱ͖ɼ
    Ω
    Ω
    Pr
    Pr(A) A
    ެཧ1
    ެཧ2
    ެཧ3
    

    View Slide

12. ඪຊۭؒͱ֬཰෼෍
    12
    αΠίϩΛճ౤͛Δͱ͖ͷඪຊۭؒ
    Ω = { , , , , , }
    ࠜݩࣄ৅
    w ඪຊۭؒ͸ࠜݩࣄ৅ͷू߹
    w ඪຊۭؒ͸ɼ΋Εͳ͘ɾͩͿΓ͕ͳ͍
    w ඪຊۭؒɹͷ෦෼ू߹ɹΛࣄ৅ʢFWFOUʣͱݺͿ
    w ɹ͸ɹͷ෦෼ू߹Ͱ΋͋Δˠશࣄ৅
    ࣄ৅ ω { } { } { } { } { } { }
    Pr(ω) 1
    6
    1
    6
    1
    6
    1
    6
    1
    6
    1
    6
    Pr
    ֬཰෼෍
    ֬཰
    ֬཰෼෍ɹɹ͸ɹɹͷ෦෼ू߹͔Β࣮਺΁ͷؔ਺
    Pr Ω
    Pr({ }) =
    1
    6
    ͷΑ͏ʹࣜͰ͔͚Δ
    Ω A
    ɹɹ
    A
    Ω
    A ⊂ Ω
    Ω Ω
    

    View Slide

13. ֬཰ͷެཧ 1
    13
    ࣄ৅Aͷ֬཰͸0Ҏ্Ͱ1ҎԼ
    0 ≤ Pr(A) ≤ 1
    

    View Slide

14. ֬཰ͷެཧ 2
    14
    શࣄ৅ͷ֬཰͸ʹ౳͍͠
    Pr(Ω) = 1
    Pr({ , , , , , }) = 1
    

    View Slide

15. ֬཰ͷެཧ 3
    15
    AͱBͷੵू߹͕ۭू߹ͳΒ͹ɼ
    AͱBͷ࿨ू߹ͷ֬཰͸ɼAͷ֬཰ͱBͷ֬཰ͷ࿨ͱ౳͍͠
    A ∩ B = ϕ Pr(A ∪ B) = Pr(A) + Pr(B)
    ͳΒ͹ɼ
    { , , , } ∩ { , , } = { , }
    A B A ∩ B
    { , , } ∩ { , , } = {}
    A B A ∩ B = ϕ
    ޓ͍ʹૉˠഉ൓ࣄ৅
    Pr({ , , , , , }) = Pr({ , , }) + Pr({ , , })
    ͳΒ͹ɼ
    ੵू߹
    

    View Slide

16. ֬཰ม਺
    16
    ֬཰ม਺͸ɼඪຊۭ͔ؒΒ࣮਺΁ͷؔ਺
    ࣄ৅ ω { } { } { } { } { } { }
    X(ω)
    X
    ֬཰ม਺
    ֬཰ม਺ͷ஋
    Pr(X(ω)) 1
    6
    1
    6
    1
    6
    1
    6
    1
    6
    1
    6
    Pr
    ֬཰෼෍
    ֬཰
    1 2 3 4 5 6
    • αΠίϩΛ1ճ౤͛ͨͱ͖ͷɼग़Δ໨Λ֬཰ม਺Xͱ͢Δ
    • ग़ͨ໨Λ100ഒͨ͠΋ͷΛ֬཰ม਺Xͱ͢Δ
    • ͋Δूஂͷ͏ͪɼநग़ͨ͠Ұਓͷ਎௕Λ֬཰ม਺Xͱ͢Δ
    • ͋Δूஂͷ͏ͪɼநग़ͨ͠Ұਓͷ͋Δ࣬පͷ༗ແΛ֬཰ม਺Yͱ͢Δ
    

    View Slide

17. ֬཰ม਺ͱ֬཰෼෍
    17
    ͋Δ࣬පͷ༗ແ ω { } { }
    x
    ֬཰ม਺ɹ͕ͱΓಘΔ஋
    Pr(X = x) 0.2 0.8
    Pr
    ֬཰෼෍
    ʹͳΔ֬཰
    1 0
    ͋Γ ͳ͠
    X
    X = x
    ؍ଌ
    ਺஋Խͨ͠؍ଌσʔλ
    

    View Slide

18. ظ଴஋Λཧղ͢Δ
    18
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

19. ظ଴஋ͷఆٛ
    19
    E[X] =
    ∞
    ∑
    k=1
    xk
    Pr(X = xk
    )
    = x1
    Pr(X = x1
    ) + x2
    Pr(X = x2
    ) + ⋯ + xn
    Pr(X = xn
    )
    ֬཰ม਺ɹ ͷظ଴஋ɹɹ Λ࣍ࣜͰఆٛ͢Δɽ͜͜Ͱɼ
    • ɹ͸ɼ֬཰ม਺ ɹ ͕ͱΔ஋
    • ɹɹɹɹ ͸ɼ֬཰ม਺ɹ ͕ɹ ʹ౳͘͠ͳΔ֬཰Λද͢
    xk
    X
    X E[X]
    Pr(X = xk
    ) X xk
    ֬཰ม਺ͷ஋ʹର͠ɼͦΕʹରԠ͢Δ֬཰ΛॏΈ෇͚͍ͯ͠Δ
    ʰྫࣔ͸ཧղͷࢼۚੴʱɹ਺ֶΨʔϧΑΓ
    

    View Slide

20. Height X n Proportion
    Probability
    150 cmͷ਎௕ͷਓ 150 10 10/40 0.25 150 * 0.25 = 37.5
    160 cmͷ਎௕ͷਓ 160 20 20/40 0.5 160 * 0.5 = 80
    170 cmͷ਎௕ͷਓ 170 10 10/40 0.25 170 * 0.25 = 42.5
    ظ଴஋ͷܭࢉ
    20
    ֬཰ม਺ɹ ͕ͱΔ஋ɹ ʹɼͦΕʹରԠ͢Δ֬཰Λ͔͚ɼશͯͷࣄ৅Λ଍͢
    xk
    X
    Pr(X = x)
    xk
    Pr(X = xk
    )
    x1
    x2
    E[X] =
    ∞
    ∑
    k=1
    xk
    Pr(X = xk
    ) = 37.5 + 80 + 42.5 = 160
    x3
    Outcome X n Proportion
    Probability
    ͋Γ 1 20 0.2 0.2 1 * 0.2 = 0.2
    ͳ͠ 0 80 0.8 0.8 0 * 0.8 = 0
    Pr(X = x)
    xk
    Pr(X = xk
    )
    x1
    x2
    E[X] =
    ∞
    ∑
    k=1
    xk
    Pr(X = xk
    ) = 0.2 + 0 = 0.2
    ࿈ଓσʔλ
    ཭ࢄσʔλʢೋ஋σʔλʣ
    xk
    Pr(X = xk
    )
    

    View Slide

21. ظ଴஋͸ฏۉʢ࿈ଓσʔλʣ
    21
    Height X n Proportion
    Probability
    150 cmͷ਎௕ͷਓ 150 10 10/40 0.25 150 * 0.25 = 37.5
    160 cmͷ਎௕ͷਓ 160 20 20/40 0.5 160 * 0.5 = 80
    170 cmͷ਎௕ͷਓ 170 10 10/40 0.25 170 * 0.25 = 42.5
    Pr(X = x)
    x1
    x2
    E[X] =
    ∞
    ∑
    k=1
    xk
    Pr(X = xk
    ) = 150 ⋅
    10
    40
    + 160 ⋅
    20
    40
    + 170 ⋅
    10
    40
    = 160
    x3
    ࿈ଓσʔλ
    xk
    Pr(X = xk
    )
    μ =
    150 ⋅ 10 + 160 ⋅ 20 + 170 ⋅ 10
    40
    = 160
    ຊདྷ͸ɼ֬཰ີ౓ؔ਺ɹɹ : Probability Density Function (PDF) Ͱߟ͑Δ
    ʢe.g. Techical Point 1.1ʣ
    E[X] =
    ∫
    x f(x)dx
    f(x)
    

    View Slide

22. ظ଴஋͸ฏۉʢ཭ࢄσʔλʣ
    22
    xk
    Pr(X = xk
    )
    Outcome X n Proportion
    Probability
    ͋Γ 1 20 0.2 0.2 1 * 0.2 = 0.2
    ͳ͠ 0 80 0.8 0.8 0 * 0.8 = 0
    Pr(X = x)
    xk
    Pr(X = xk
    )
    x1
    x2
    ཭ࢄσʔλʢೋ஋σʔλʣ
    E[X] =
    ∞
    ∑
    k=1
    xk
    Pr(X = xk
    ) = 1 ⋅
    20
    100
    + 0 ⋅
    80
    100
    = 0.2 μ =
    1 ⋅ 20 + 0 ⋅ 80
    100
    = 0.2
    • ಛʹೋ஋σʔλͷ৔߹͸ɼ
    ظ଴஋ = ฏۉ = ֬཰ม਺͕1ʹͳΔׂ߹
    • ൃ঱ͷ༗ແΛ1ͱ0ʹίʔσΟϯάͨ͠৔߹ɼൃ঱ׂ߹
    ಛʹɼೋ஋σʔλΛ1, 0Ͱࣔ֬͢཰ม਺ΛIndicatorͱݺͿ
    

    View Slide

23. ظ଴஋ͷઢܗੑ (1)
    23
    ֬཰ม਺ɹɹɹͷظ଴஋ɹɹɹɹ ʹ͍ͭͯɼ͕࣍ࣜ੒Γཱͭ
    X + Y E[X + Y]
    E[X + Y] = E[X] + E[Y]
    ࿨ͷظ଴஋͸ɼظ଴஋ͷ࿨ xk
    Pr(X = xk
    )
    ID X Y X + Y
    1 1 1 2
    2 1 0 1
    3 1 1 2
    4 1 1 2
    5 1 0 1
    6 1 0 1
    7 1 0 1
    8 1 0 1
    9 0 1 1
    10 0 0 0
    X + Y Probability
    2 3/10 2 * 0.3 = 0.6
    1 6/10 1 * 0.6 = 0.6
    0 1/10 0 * 0.1 = 0
    (x + y)k
    ⋅ Pr(X + Y)
    E[X] = 0.8, E[Y] = 0.4 E[X] + E[Y] = 1.2
    E[X + Y] = 1.2
    

    View Slide

24. ظ଴஋ͷઢܗੑ (2)
    24
    E[k ⋅ X] = k ⋅ E[X]
    ·ͨɼ೚ҙͷఆ਺ɹʹ͍ͭͯɼ͕࣍ࣜ੒Γཱͭ
    ఆ਺ഒͷظ଴஋͸ɼظ଴஋ͷఆ਺ഒ
    X
    (Height [cm])
    k
    kX
    (Height [m])
    Probability
    150 1/100 1.5 0.25 150 * 0.25 = 37.5 0.375
    160 1/100 1.6 0.5 160 * 0.5 = 80 0.80
    170 1/100 1.7 0.25 170 * 0.25 = 42.5 0.425
    xk
    Pr(X = xk
    ) kxk
    Pr(X = kxk
    )
    E[X] = 160
    kE[X] = 1/100 ⋅ 160
    = 1.6
    E[kX] = 1.6
    k
    x1
    x2
    x3
    

    View Slide

25. Average causal effect ΛಡΉ
    25
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y1
    ] − E[Y2
    ] = E[Y1
    − Y2
    ]
    ID Y1
    Y2
    Y1
    - Y2
    1 1 1 0
    2 1 0 1
    3 1 1 0
    4 1 1 0
    5 1 0 1
    6 1 0 1
    7 1 0 1
    8 1 0 1
    9 0 1 -1
    10 0 0 0
    Y1
    - Y2 Probability
    -1 1/10 -0.1
    0 4/10 0
    1 5/10 0.5
    (y1
    − y2
    )k
    ⋅ Pr(Y1
    − Y2
    )
    E[Y1
    ] = 0.8, E[Y2
    ] = 0.4
    E[Y1
    ] − E[Y2
    ] = 0.4
    E[Y1
    − Y2
    ] = 0.4
    • ظ଴஋ͷઢܗੑΑΓɼಉ͡஋ʹͳΔ
    • ࠨล͸ɼ͋Δ2ͭͷঢ়ଶʹ͓͚Δɼ
    ͦΕͧΕͷΞ΢τΧϜͷظ଴஋ͷࠩ
    • ӈล͸ɼݸਓ಺Ͱͷ2ͭͷঢ়ଶʹ͓͚Δ
    Ξ΢τΧϜͷࠩͷظ଴஋
    ղऍ͠΍͍͢
    

    View Slide

26. ৚͖݅ͭظ଴஋Λ෼ղ͢Δ
    26
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

27. ৚͖݅ͭ֬཰ɼಉ࣌֬཰
    27
    P(Y = y|A = a)
    ৚͖݅ͭ֬཰ɿɹɹɹͰϑΟϧλʔΛ͔͚ͨޙͷɹɹɹͷ֬཰
    A = a Y = y
    k
    Outcome
    Y
    Sex
    A
    n
    Joint probability
    1 1 (M) 20 20/200 = 0.1
    1 0 (F) 50 50/200 = 0.25
    0 1 (M) 80 80/200 = 0.4
    0 0 (F) 50 50/200 = 0.25
    Pr(Y, A)
    ಉ࣌֬཰ɿ
    ෳ਺ͷ֬཰ม਺͕ಉ࣌ʹى͜Δ֬཰
    k
    Outcome
    Y
    Sex
    A = 1
    n
    Conditional probability
    1 1 20 20/100 = 0.2
    0 1 80 80/100 = 0.8
    Pr(Y |A = 1)
    k
    Outcome
    Y
    Sex
    A = 0
    n
    Conditional probability
    1 0 50 50/100 = 0.5
    0 0 50 50/100 = 0.5
    Pr(Y |A = 0)
    αϒάϧʔϓ಺
    

    View Slide

28. ৚͖݅ͭظ଴஋
    28
    E[Y|A = a] =
    ∞
    ∑
    k=1
    yk
    Pr(Y = yk
    |A = a)
    = y1
    Pr(Y = y1
    |A = a) + y2
    Pr(Y = y2
    |A = a) + ⋯ + yn
    Pr(Y = yn
    |A = a)
    Outcome
    Y
    Sex
    A = 1
    n
    Conditional probability
    1 1 20 20/100 = 0.2 1 * 0.2 = 0.2
    0 1 80 80/100 = 0.8 0 * 0.8 = 0
    Outcome
    Y
    Sex
    A = 0
    n
    Conditional probability
    1 0 50 50/100 = 0.5 1 * 0.5 = 0.5
    0 0 50 50/100 = 0.5 0 * 0.5 = 0.5
    yk
    Pr(Y = yk
    |A = 1)
    yk
    Pr(Y = yk
    |A = 0)
    Pr(Y |A = 0)
    Pr(Y |A = 1)
    E[Y|A = 1] = 0.2
    E[Y|A = 1] = 0.5
    ੑผͰαϒάϧʔϓʹ෼͚ͨޙɼαϒάϧʔϓ಺Ͱͷظ଴஋ΛٻΊΔ
    

    View Slide

29. Association measure ΛಡΉ
    29
    ID
    Outcome
    Y
    Exposure
    A
    1 1 1
    2 0 1
    3 0 1
    4 0 1
    5 1 1
    6 0 0
    7 1 0
    8 1 0
    9 1 0
    10 0 0
    E[Y|A = 1] − E[Y|A = 0]
    Outcome
    Y
    Exposure
    A = 1
    n
    Conditional probability
    1 1 2 2/5 = 0.4
    0 1 3 3/5 = 0.6
    yk
    Pr(Y = yk
    |A = 1)
    Pr(Y |A = 1)
    E[Y|A = 1] = 0.4
    Outcome
    Y
    Exposure
    A = 0
    n
    Conditional probability
    1 0 3 3/5 = 0.6
    0 0 2 2/5 = 0.4
    yk
    Pr(Y = yk
    |A = 1)
    Pr(Y |A = 0)
    E[Y|A = 0] = 0.6
    E[Y|A = 1] − E[Y|A = 0] = 0.4 − 0.6 = − 0.2
    മ࿐܈ͷ࣬ױൃੜͷظ଴஋ͱඇമ࿐܈ͷ࣬ױൃੜͷظ଴஋ͷࠩ
    

    View Slide

30. ৚͖݅ͭظ଴஋ͱճؼ (1)
    30
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

31. ৚͖݅ͭظ଴஋ͱճؼ (2)
    31
    ճؼͱ͸৚͖݅ͭظ଴஋ΛٻΊΔ͜ͱ
    ৚͖݅ͭظ଴஋͸ɼճؼͰٻΊΒΕΔ
    E[Y|A = a]
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    160
    165
    170
    175
    180
    12 14 16
    Age [yr]
    Height [cm]
    E[Height|Age] E[Height|Age, sex = f ]
    E[Height|Age, sex = m]
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ●
    ● ●
    ●
    ●
    ●
    160
    165
    170
    175
    180
    12 14 16
    Age [yr]
    Height [cm]
    sex
    ● f
    m
    

    View Slide

32. ஌ͬͱ͖͍ͨςΫχοΫ (1)
    32
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

33. ৚͖݅ͭ֬཰
    33
    Pr(Y|A) =
    Pr(Y, A)
    Pr(A)
    k
    Outcome
    Y
    Sex
    A
    n
    Joint probability Conditional probability
    1 1 20 20/200 = 0.1 0.1/0.5 = 0.2
    1 0 50 50/200 = 0.25 0.25/0.5 = 0.5
    0 1 80 80/200 = 0.4 0.4/0.5 = 0.8
    0 0 50 50/200 = 0.25 0.25/0.5 = 0.5
    Pr(Y, A)
    Pr(A = 1) = (20 + 80)/200 = 0.5
    Pr(Y, A) Pr(Y |A = a)
    Pr(A = 0) = (50 + 50)/200 = 0.5
    

    View Slide

34. Chain rule (1)
    34
    Pr(Y, A) = Pr(Y|A) ⋅ Pr(A)
    ৐๏ఆཧ
    ಉ࣌֬཰͔Βɼ৚͚͍݅ͮͨม਺Λు͖ग़͢
    Oݸͷࣄ৅ʹ͍ͭͯ΋֦ுͰ͖Δ
    $IBJOSVMF Pr(A1
    , A2
    , ⋯, An
    ) = Pr(An
    |A1
    , A2
    , ⋯, An−1
    )⋯ Pr(A2
    |A1
    ) Pr(A1
    )
    ಉ࣌֬཰͔Βɼ৚͚͍݅ͮͨม਺Λు͖ग़͠ଓ͚Δ
    Pr(Y|A) =
    Pr(Y, A)
    Pr(A)
    

    View Slide

35. Chain rule (2)
    35
    $IBJOSVMF Pr(A1
    , A2
    , ⋯, An
    ) = Pr(An
    |A1
    , A2
    , ⋯, An−1
    )⋯ Pr(A2
    |A1
    ) Pr(A1
    )
    ʰྫࣔ͸ཧղͷࢼۚੴʱɹ਺ֶΨʔϧΑΓ
    Pr(A1
    , A2
    , A3
    ) = Pr(A1
    |A2
    , A3
    ) Pr(A2
    , A3
    )
    = Pr(A1
    |A2
    , A3
    ) Pr(A2
    |A3
    ) Pr(A3
    )
    Pr(A1
    , A2
    , A3
    , A4
    ) = Pr(A1
    |A2
    , A3
    , A4
    ) Pr(A2
    , A3
    , A4
    )
    = Pr(A1
    |A2
    , A3
    , A4
    ) Pr(A2
    |A3
    , A4
    ) Pr(A3
    , A4
    )
    = Pr(A1
    |A2
    , A3
    , A4
    ) Pr(A2
    |A3
    , A4
    ) Pr(A3
    |A4
    ) Pr(A4
    )
    Pr(A1
    , A2
    , A3
    ) = Pr(A2
    |A1
    , A3
    ) Pr(A1
    , A3
    )
    = Pr(A2
    |A1
    , A3
    ) Pr(A1
    |A3
    ) Pr(A3
    )
    ೚ҙͷࣄ৅Λు͖ग़ͤΔ
    

    View Slide

36. ৐๏ఆཧ͔Β৚͖݅ͭظ଴஋΁
    36
    Pr(Y, A) = Pr(Y|A) ⋅ Pr(A)
    E[Y, A] = E[Y|A] ⋅ Pr(A)
    ॏΈ͚ͮ
    ඪ४ԽͰ࢖͏
    Outcome
    Y
    Exposure
    A = 1
    n
    Conditional probability
    1 1 2 2/5 = 0.4
    0 1 3 3/5 = 0.6
    yk
    Pr(Y = yk
    |A = 1)
    Pr(Y |A = 1)
    E[Y|A = 1] = 0.4
    Outcome
    Y
    Exposure
    A = 0
    n
    Conditional probability
    1 0 3 3/5 = 0.6
    0 0 2 2/5 = 0.4
    yk
    Pr(Y = yk
    |A = 1)
    Pr(Y |A = 0)
    E[Y|A = 0] = 0.6
    E[Y, A = 1] = 0.4 ⋅ 0.5 = 0.2
    E[Y, A = 0] = 0.6 ⋅ 0.5 = 0.3
    Pr(A = 1) = 0.5
    Pr(A = 0) = 0.5
    

    View Slide

37. ಠཱੑ
    37
    Pr(Y|A) = Pr(Y)
    Pr(Y, A)
    Pr(A|Y) = Pr(A)
    Pr(Y, A) = Pr(Y) ⋅ Pr(A)
    2ͭͷࣄ৅YͱAʹ͍ͭͯɼ͕࣍ࣜ੒Γཱͭͱ͖ɼࣄ৅YͱA͸ಠཱͰ͋Δ
    ಠཱͰ͋Δͱ͖ɼ͕࣍ࣜಘΒΕΔ
    Y⊥
    ⊥ A ͱ͔͘
    keynoteͷtex
    Ͱॻ͚ͨΑ!!
    ExchangeabilityͰ࢖͏
    հೖAΛϥϯμϜʹׂΓ෇͚ͯ΋ɼજࡏΞ΢τΧϜʹ͸Өڹ͠ͳ͍
    

    View Slide

38. ৚͖݅ͭಠཱ
    38
    Pr(Y|A, L) = Pr(Y|L)
    Pr(Y, A)
    Pr(Y, A|L) = Pr(Y|L) ⋅ Pr(A|L)
    3ͭͷࣄ৅YɼAɼLʹ͍ͭͯɼ͕࣍ࣜ੒Γཱͭͱ͖ɼ
    LΛ৚্͚݅ͮͨͰࣄ৅YͱA͸ಠཱͰ͋Δ
    ৚͖݅ͭಠཱͰ͋Δͱ͖ɼ͕࣍ࣜಘΒΕΔ
    Y⊥
    ⊥ A|L ͱ͔͘
    Conditional ExchangeabilityͰ࢖͏
    αϒάϧʔϓ಺Ͱ͸ɼհೖA͸ϥϯμϜʹׂΓ෇͚ΒΕ͍ͯΔͱߟ͑Δ
    ͢ΔͱհೖA͸જࡏΞ΢τΧϜʹ͸Өڹ͠ͳ͍
    

    View Slide

39. DAGΛ਺ࣜͰද͢
    39
    w Chain ruleͱಠཱੑ͔ΒDAGΛ਺ࣜͰදݱͰ͖Δ
    w άϥϑΟΧϧϞσϧʢஞ࣍తҼ਺෼ղͷ๏ଇʣ͔Βͷ੍໿͸ඞཁ
    • ैଐؔ܎ΛChain ruleʹ෇༩͢Δ
    Y
    A L Pr(Y, A, L) = Pr(Y|L) Pr(L|A) Pr(A)
    A Y Pr(Y, A) = Pr(Y|A) Pr(A)
    A Y Pr(Y, A) = Pr(Y) Pr(A)
    Y
    A
    L Pr(Y, A, L) = Pr(Y|A, L) Pr(A|L) Pr(L)
    Pr(Y, A, L) = Pr(Y|A, L) Pr(L, A) Pr(A)
    = Pr(Y|A, L) Pr(L|A) Pr(A)
    L
    Y
    A Pr(Y, A, L) = Pr(L|Y, A) Pr(Y) Pr(A)
    

    View Slide

40. ஌ͬͱ͖͍ͨςΫχοΫ (2)
    40
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

41. શ֬཰ͷެࣜͱपลԽ
    41
    ࣄ৅AͱB͕͋Δɽࣄ৅Bͷཁૉಉ࢜΋ഉ൓ͳΒ͹ɼ͕࣍ࣜ੒Γཱͭ
    Pr(A) = Pr(A, B1
    ) + Pr(A, B2
    ) + ⋯ + Pr(A, Bn
    )
    Probability Exposure
    Outcome A = 1 A = 0
    Y = 1 0.1 0.25 0.35
    Y = 0 0.4 0.25 0.65
    0.5 0.5 1
    Pr(Y)
    Pr(A)
    Pr(Y, A)
    Pr(A = 1) = Pr(A = 1,Y = 1) + Pr(A = 1,Y = 0)
    = 0.1 + 0.4 = 0.5
    Pr(A = 0) = Pr(A = 0,Y = 1) + Pr(A = 0,Y = 0)
    = 0.25 + 0.25 = 0.5
    Pr(Y = 1) = Pr(Y = 1,A = 1) + Pr(Y = 1,A = 0)
    = 0.1 + 0.25 = 0.35
    Pr(Y = 0) = Pr(Y = 0,A = 1) + Pr(Y = 0,A = 0)
    = 0.4 + 0.25 = 0.65
    पล֬཰
    पล֬཰ΛٻΊΔ͜ͱΛ
    पลԽͱݺͿ
    

    View Slide

42. ৚͖݅ͭ֬཰ͱपลԽ
    42
    શ֬཰ͷެࣜ͸ɼ৚͖݅ͭ֬཰Ͱද͢͜ͱ͕Ͱ͖Δ
    Pr(A) = Pr(A, B1
    ) + Pr(A, B2
    ) + ⋯ + Pr(A, Bn
    )
    = Pr(A|B1
    ) Pr(B1
    ) + Pr(A|B2
    ) Pr(B2
    ) + ⋯ + Pr(A|Bn
    ) Pr(Bn
    )
    Probability Exposure
    Outcome A = 1 A = 0
    Y = 1 0.2 0.5 0.2 * 0.5 = 0.1 0.5 * 0.5 = 0.25 0.35
    Y = 0 0.8 0.5 0.8 * 0.5 = 0.4 0.5 * 0.5 = 0.25 0.65
    0.5 0.5 1
    Pr(Y |A = 1) Pr(A = 1)
    Pr(A)
    Pr(Y |A)
    മ࿐AͰ৚͚݅ͭΔ
    Pr(Y |A = 0) Pr(A = 0) Pr(Y)
    पล֬཰
    ඪ४ԽͰ࢖͏
    E[Y] = E[Y |A1
    ] ⋅ P(A1
    ) + E[Y |A2
    ] ⋅ P(A2
    ) + ⋯ + E[Y |An
    ] ⋅ P(An
    )
    =
    n
    ∑
    k=1
    E[Y |Ak
    ] ⋅ Pr(Ak
    )
    

    View Slide

43. Roadmap
    43
    Average causal effect ͱ Association measureͷ ߏ੒ཁૉΛཧղ͢Δ
    Goal
    Average causal effect
    Association measure
    E[Ya=1] − E[Ya=0] = E[Ya=1 − Ya=0]
    E[Y|A = 1] − E[Y|A = 0]
    ظ଴஋ ظ଴஋ͷઢܗੑ
    ฏۉ ֬཰
    ֬཰ม਺
    ৚͖݅ͭظ଴஋
    ৚͖݅ͭ֬཰
    Marginal
    Conditional
    w ಉ࣌֬཰
    w ಠཱੑ
    w $IBJOSVMF
    w શ֬཰ͷެࣜ
    पลԽ
    ճؼ
    पลԽ
    

    View Slide

44. ·ͱΊʢ͜Ε͚ͩ͸֮͑ͯʂʣ
    44
    1. ڵຯ͋Δม਺ͷظ଴஋ = ڵຯ͋Δม਺ͷฏۉ
    ɹೋ஋σʔλͷ৔߹ɼظ଴஋ = ฏۉ = ׂ߹
    2. ʰ࿨ͷظ଴஋͸ɼظ଴஋ͷ࿨ʱʢ਺ֶΨʔϧ ཚ୒ΞϧΰϦζϜΑΓʣ
    3. ৚͖݅ͭظ଴஋͸ɼαϒάϧʔϓ಺Ͱͷظ଴஋
    E[X + Y] = E[X] + E[Y]
    E[Y|A = 1]
    E[Y|A = 0]
    Y A
    1
    1
    1
    0
    0
    0
    

    View Slide

45. • Hernán MA, Robins JM. (2020). Causal Inference: What If. Boca Raton: Chapman
    & Hall/CRC.
    • Pearl J, (མւߒ, ༁). (2019). ౷ܭతҼՌਪ࿦. ே૔ॻళ.
    • ౻ᖒ༸ಙ. (2006). ֬཰ͱ౷ܭ. ே૔ॻళ.
    • A. ίϧϞΰϩϑ, I. δϡϧϕϯί, A. ϓϩϗϩϑ, ؙࢁ఩࿠+അ৔ྑ࿨. (2003).
    ίϧϞΰϩϑͷ֬཰࿦ೖ໳. ৿๺ग़൛.
    • स໦௚ٱ. (2004). ֬཰࿦. ே૔ॻళ.
    • ݁৓ߒ. (2011). ਺ֶΨʔϧ ཚ୒ΞϧΰϦζϜ. SB Creative.
    • ݁৓ߒ. (2016). ਺ֶΨʔϧͷൿີͷϊʔτ ΍͍͞͠౷ܭ. SB Creative.
    • ࠇ໦ֶ. (2020). ਺ཧ౷ܭֶ ౷ܭతਪ࿦ͷجૅ. ڞཱग़൛.
    • Porta M. (2014). A dictionary of epidemiology sixth edition. Oxford.
    ࢀߟจݙ
    45
    

    View Slide