Identification of Causal Effect in the Presence of Selection Bias - - PowerPoint PPT Presentation

Identification of Causal Effect in the Presence of Selection Bias

Juan D. Correa Jin Tian Elias Bareinboim

AAAI Honolulu, 2019

Challenge 1: Confounding Bias

Exercise (Hours) Cholesterol

Age Exercise Cholesterol What’s the causal effect of Exercise on Cholesterol? What about 𝑄 𝑑ℎ𝑝𝑚𝑓𝑡𝑢𝑓𝑠𝑝𝑚 𝑓𝑦𝑓𝑠𝑑𝑗𝑡𝑓) ?

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Challenge 1: Confounding Bias

Exercise (Hours) Cholesterol

Age 10 Age 20 Age 30 Age 40 Age 50 Age Exercise Cholesterol

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Challenge 1: Confounding Bias

Exercise (Hours) Cholesterol

Age 10 Age 20 Age 30 Age 40 Age 50 Age Exercise Cholesterol

𝑄 𝑑ℎ𝑝𝑚𝑓𝑡𝑢𝑓𝑠𝑝𝑚 𝑒𝑝(𝑓𝑦𝑓𝑠𝑑𝑗𝑡𝑓)) ≠ 𝑄(𝑑ℎ𝑝𝑚𝑓𝑡𝑢𝑓𝑠𝑝𝑚 | 𝑓𝑦𝑓𝑠𝑑𝑗𝑡𝑓)

This difference is called Confounding Bias

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Challenge 2: Selection Bias

Exercise (Hours) Cholesterol

S=1 S=0 Variables in the system affect the inclusion of units in the sample Age Exercise Cholesterol S Fitness

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Challenge 2: Selection Bias

Exercise (Hours) Cholesterol

S=1 S=0 Variables in the system affect the inclusion of units in the sample Age Exercise Cholesterol S Fitness

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𝑄(𝑏𝑕𝑓, 𝑓𝑦, 𝑑ℎ, 𝑔𝑗𝑢) ≠ 𝑄 𝑏𝑕𝑓, 𝑓𝑦, 𝑑ℎ, 𝑔𝑗𝑢 𝑇 = 1)

This difference is due to Selection Bias

Current literature

Association = Causation No control

Complete Algorithms

[Tian and Pearl ’02; Huang and Valtorta ’06; Shpitser and Pearl ’06; Bareinboim and Pearl ’12] Controlling Selection Bias [Bareinboim and Pearl ’12] Recovering from Selection Bias in Causal and Statistical Inference [Bareinboim, Tian, Pearl ’14] RCE [Bareinboim, Tian, Pearl ’15] Generalized Adjustment

[Correa, Tian, Bareinboim ’18]

IDSB

[Correa, Tian, Bareinboim ’19]

No Confounding Confounding No Selection Selection

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Problem I

Given: Is there a function 𝑔 such that

𝑄 𝒛 𝑒𝑝 𝒚 = 𝑔(𝑄

;)

?

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𝒣

𝑇 𝑄(𝒘|𝑇 = 1)

1 … 1 … 1 …

𝑄

Variables

𝒀, 𝒁

Result 1

Theorem 1: Let 𝒀, 𝒁 ⊂ 𝑾 be two disjoint sets of variables and 𝒣 a causal diagram

• ver 𝑾 and 𝑇. If 𝒁 ⊥ 𝑇 𝒣𝒀𝒁

CDE, then 𝑄

𝒚(𝒛) is not recoverable from

𝑄(𝒘 | 𝑇 = 1) in 𝒣.

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Problem II

Given: Is there a function 𝑔 such that

𝑄 𝒛 𝑒𝑝 𝒚 = 𝑔(𝑄

;, 𝑄F)

?

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𝒣

𝑇 𝑄(𝒘|𝑇 = 1)

1 … 1 … 1 …

𝑄(𝒖)

… … …

𝑄

;

𝑄F

Variables

𝒀, 𝒁

Result II

Algorithm IDSB Given a causal diagram, a selection-biased distribution and external data

• ver a subset of the variables and the variables of interest (𝒀, 𝒁);

returns an expression for 𝑄

𝒚(𝒛) in terms of the input or failure.

Strictly more powerful than the best known algorithm that accepts both biased and unbiased data.

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Decomposing the Problem

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X W1 W2 W3 Y S

Intervention

𝑄

H 𝑧 =

J 𝑄

H(𝑧, 𝑥L, 𝑥F, 𝑥;)

• NO,NP,NQ

X W1 W2 W3 Y S

Decomposing the Problem

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W1 W3 S

C-Components

W2 Y

𝑄

H 𝑧 =

J 𝑄

H(𝑧, 𝑥L, 𝑥F, 𝑥;)

• NO,NP,NQ

= J 𝑄

H,NO,NQ 𝑧, 𝑥F 𝑄 NP,R 𝑥L, 𝑥;

• NO,NP,NQ

X W1 W2 W3 Y S

𝑄

H,NO,NQ 𝑧, 𝑥F

𝑄

NP,R 𝑥L, 𝑥;

Summary

• 1. Complete characterization recoverable causal effects from

the causal diagram and a selection-biased probability distribution.

• 2. Sufficient procedure to recover causal effects from a causal

diagram, selection-biased distributions and auxiliary unbiased data which is strictly more powerful than state-of- the-art procedure.

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Thanks!

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