Shuntaro Sato
April 27, 2020
4.6k

# 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．

April 27, 2020

## Transcript

1. Shuntaro Sato
([email protected]ੜ෺౷ܭՈ)
ӸֶͷͨΊͷ֬཰ͷجૅ
Causal Inference: What Ifษڧձ

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

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

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

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

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

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

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

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

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཰

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

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

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

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

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
ੵू߹

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

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

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

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

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
)

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)

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

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

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

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΍͍͢

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

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಺

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

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

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

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

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

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

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)

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

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

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

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

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)

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

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

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
)

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

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

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