Causal Model Ying Nian Wu UCLA Department of Statistics July 13, 2007 IPAM Summer School

Causal Model

Ying Nian Wu

UCLA Department of Statistics

July 13, 2007IPAM Summer School

Observational study --- observed relationship may not be cause-effect

Example: people who sleep 7 hours report better health

sleep 7 hrs (vs 8 hrs)

health

health

sleep 7 hrs (vs 8 hrs)

Example: people who smoke cigarette have better health than people who smoke pipe

cigarette (vs pipe)

health

cigarette (vs pipe)

health

age

cigarette (vs pipe) health

Confounding variable

Donald B. Rubin EM algorithm – Dempster, Laird, Rubin Missing data: ignorability multiple imputation Little & Rubin book Bayesian statistics: foundations and applications Gelman et al. book Causality: Rubin causal model Neyman-Rubin model

Rubin’s potential outcomeCounterfactual intervention

sleep 7 hrs (vs 8 hrs)

health

e.g., what would have happen had the same person who sleeps7 hrs slept 8 hrs instead?

Rubin’s potential outcomeCounterfactual intervention

cigarette (vs pipe)

health

e.g., what would have happen had the same person who smokespipe smoked cigarette instead?

Rubin’s advice

Define estimand before trying to estimate it from data.

Counterfactual intervention:

why counterfactual? we cannot jump into the same river twice fundamentally missing data problem

define estimand in terms of complete data try to estimate it in the presence of missing data

Experiment: randomized assignment or interventionObservational study: actual intervention not ethical

Today’s reference is Judea Pearl, Causality

What is a causal model and what it can do for us?How to learn a causal model, structure and parameters?

Cochran example

0Z

B

X

Y

1Z

2Z 3Z

X

Y

0Z

1Z

2Z

3Z

BCausal diagram

Soil fumigant

Oat crop yieldsEelworm populationZ

Last year -- unobserved

Before treatment

After treatment

End of season

Birds -- unobserved

0Z

B

X

Y

1Z

2Z 3Z

X

Y

Soil fumigant

Oat crop yieldsEelworm populationZ

Farmers insist on they decide X ,which depends on 0ZHow to define causal effect of X on Y ?

Can it be obtained from passive observations?

Causal Model

0Z

B

X

Y

1Z

2Z 3Z

X

Y

Soil fumigant

Oat crop yieldsEelworm populationZ

Causal diagram: more than conditional independence

0Z

B

X

Y

1Z

2Z 3Z

Causal Model

Causal diagram

)( 000 fZ

),( 1011 ZfZ

),,( 2122 ZXfZ

),,( 3233 ZBfZ

),( 0 BB ZfB

),( 0 XX ZfX

),,,( 32 YY ZZXfY

Structural equations

’s are independent

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B

X

Y

1Z

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Rubin’s potential outcome

),,( 2122 ZXfZ

Counterfactual intervention

equationsor ablesother vari of regardless

),,,( bemust then

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11

zxfZ

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Non-experimental observationsRepeat 1 million times

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),,( 2122 ZXfZ

),,( 3233 ZBfZ

),( 0 BB ZfB

),( 0 XX ZfX

),,,( 32 YY ZZXfY return ),,,,( 321 YZZZX

End

Get a new set of 0Z

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X

Y

1Z

2Z 3Z

A million copies of

),,,,( 321 YZZZX

knownblack

Causal effect: intervention

Repeat 1 million times

)( 000 fZ

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),,( 2122 ZXfZ

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),( 0 BB ZfB

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),,,( 32 YY ZZXfY End

Get a new set of 0Z

B

X

Y

1Z

2Z 3Z

A distribution of

black

Y

xX

)at set |Pr( xXyY

black

My codeobservingmode

My codeinterveningmode

You guess

Let’s play a game

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at thoselook about what

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Not a million ),,,,,,( 3210 YZZZXZB

Causal effect may not be identifiable from observational study

But can we express )|Pr( xXyY without ?,0 bz

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You guess

What is a causal model and what it can do for us?How to learn a causal model, structure and parameters?