1 reasoning with causes and counterfactuals judea pearl ucla (judea/)

REASONING WITH CAUSESAND COUNTERFACTUALS

Judea PearlUCLA

(www.cs.ucla.edu/~judea/)

1. Unified conceptualization of counterfactuals,

structural-equations, and graphs

2. Confounding demystified

3. Direct and indirect effects (Mediation)

4. Transportability and external validity mathematized

OUTLINE

TRADITIONAL STATISTICALINFERENCE PARADIGM

Inference

Q(P)(Aspects of P)

PJoint

Distribution

e.g.,Infer whether customers who bought product Awould also buy product B.Q = P(B | A)

Inference

Q(M)(Aspects of M)

Data Generating

M – Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis.

JointDistribution

THE STRUCTURAL MODELPARADIGM

“Think Nature, not experiment!”•

INPUT OUTPUT

FAMILIAR CAUSAL MODELORACLE FOR MANIPILATION

Y := 2X

Correct notation:

Y = 2X

WHY PHYSICS DESERVESA NEW ALGEBRA

Had X been 3, Y would be 6.If we raise X to 3, Y would be 6.Must “wipe out” X = 1.

X = 1 Y = 2

The solutionProcess informationX = 1

e.g., Length (Y) equals a constant (2) times the weight (X)

Scientific Equations (e.g., Hooke’s Law) are non-algebraic

STRUCTURALCAUSAL MODELS

Definition: A structural causal model is a 4-tupleV,U, F, P(u), where• V = {V1,...,Vn} are endogeneas variables• U = {U1,...,Um} are background variables• F = {f1,..., fn} are functions determining V,

vi = fi(v, u)• P(u) is a distribution over UP(u) and F induce a distribution P(v) over observable variables

Yuxy e.g.,

CAUSAL MODELS AND COUNTERFACTUALS

Definition: The sentence: “Y would be y (in situation u), had X been x,”

denoted Yx(u) = y, means:The solution for Y in a mutilated model Mx, (i.e., the equations for X

replaced by X = x) with input U=u, is equal to y.

)()( uYuY xMx

The Fundamental Equation of Counterfactuals:

READING COUNTERFACTUALSFROM SEM

Data shows: = 0.7, = 0.5, = 0.4A student named Joe, measured X=0.5, Z=1, Y=1.9Q1: What would Joe’s score be had he doubled his study time?

X Y7.0

5.0 4.03

TimeStudy

Treatment

9.1)(2 uYz

Q1: What would Joe’s score be had he doubled his study time?Answer: Joe’s score would be 1.9Or,In counterfactual notation:

1 5.04.0

READING COUNTERFACTUALS

X Y7.0

5.0 4.03

5.0X 5.1

1X 0 25.1

5.0X Y 25.2

4.075.03

Q2: What would Joe’s score be, had the treatment been 0 and had he studied at whatever level he would have studied had the treatment been 1?

READING COUNTERFACTUALS

X Y7.0

5.0 4.03

Z 0.2Z

Y 9.17.0

1 5.04.0

5.0X 5.1

25.1Z4.0

1 25.2

POTENTIAL AND OBSERVED OUTCOMES PREDICTED BY A STRUCTURAL MODEL

In particular:

),|(),|'(

)()()|(

')(':'

yxuPyxyYP

uPyYPyP

yuxYux

CAUSAL MODELS AND COUNTERFACTUALS

Definition: The sentence: “Y would be y (in situation u), had X been x,”

denoted Yx(u) = y, means:

The solution for Y in a mutilated model Mx, (i.e., the equations for X replaced by X = x) with input U=u, is equal to y.

)(),()(,)(:

uPzZyYPzuZyuYu

Joint probabilities of counterfactuals:

Define:

Assume:

Identify:

Estimate:

THE FIVE NECESSARY STEPSOF CAUSAL ANALYSIS

Express the target quantity Q as a function Q(M) that can be computed from any model M.

Formulate causal assumptions A using some formal language.

Determine if Q is identifiable given A.

Estimate Q if it is identifiable; approximate it, if it is not.

Test the testable implications of A (if any).

))(|())(|( 01 xdoYExdoYE ATEe.g., )'|( xXyYP x ETT e.g.,

),(|),,( 010001 xxZxxxZx YYEIEYYEDE e.g., )'|( xXyYP x ETT e.g.,

Express the target quantity Q as a function Q(M) that can be computed from any model M.

Formulate causal assumptions A using some formal language.

Determine if Q is identifiable given A.

Estimate Q if it is identifiable; approximate it, if it is not.

Test the testable implications of A (if any).

),|( 1100yYxXyYPPC x e.g.,

Define:

Assume:

Identify:

Estimate:

THE FIVE NECESSARY STEPSOF CAUSAL ANALYSIS

CAUSAL MODEL

A - CAUSAL ASSUMPTIONS

Q Queries of interest

Q(P) - Identified estimands

Data (D)

Q - Estimates of Q(P)

Causal inference

T(MA) - Testable implications

Statistical inference

Goodness of fit

Model testingProvisional claims

),|( ADQQ )(Tg

A* - Logicalimplications of A

CAUSAL MODEL

THE LOGIC OF CAUSAL ANALYSIS

IDENTIFICATION IN SCM

Find the effect of X on Y, P(y|do(x)), given the

causal assumptions shown in G, where Z1,..., Zk are auxiliary variables.

Can P(y|do(x)) be estimated if only a subset, Z, can be measured?

ELIMINATING CONFOUNDING BIASTHE BACK-DOOR CRITERION

P(y | do(x)) is estimable if there is a set Z ofvariables such that Z d-separates X from Y in Gx.

• Moreover,

(“adjusting” for Z) Ignorability

z z zxP

zyxPzPzxyPxdoyP

)|(),,(

)(),|())(|(

Front Door

EFFECT OF WARM-UP ON INJURY (After Shrier & Platt, 2008)

No, no!

Watch out!

Warm-up Exercises (X) Injury (Y)

PROPENSITY SCORE ESTIMATOR(Rosenbaum & Rubin, 1983)

),,,,|1(),,,,( 5432154321 zzzzzXPzzzzzL

Adjustment for L replaces Adjustment for Z

lLPxlLyPzPxzyP )(),|()(),|(Theorem:

P(y | do(x)) = ?

Can L replace {Z1, Z2, Z3, Z4, Z5} ?

WHAT PROPENSITY SCORE (PS)PRACTITIONERS NEED TO KNOW

1. The asymptotic bias of PS is EQUAL to that of ordinary adjustment (for same Z).

2. Including an additional covariate in the analysis CAN SPOIL the bias-reduction potential of others.

)|1()( zZXPzL

lPxlyPzPxzyP )(),|()(),|(

3. In particular, instrumental variables tend to amplify bias.4. Choosing sufficient set for PS, requires knowledge of the

model.

ZX YX Y X Y

SURPRISING RESULT:Instrumental variables are Bias-Amplifiers in linear models (Bhattarcharya & Vogt 2007; Wooldridge 2009)

“Naive” bias

Adjusted bias

11))(|(),|( cc

cccxdoYE

2102100 ))(|()|( ccccccxdoYEx

(Unobserved)

INTUTION:When Z is allowed to vary, it absorbs (or explains) some of the changes in X.

When Z is fixed the burden falls on U alone, and transmitted to Y (resulting in a higher bias)

WHAT’S BETWEEN AN INSTRUMENT AND A CONFOUNDER?Should we adjust for Z?

ANSWER:

CONCLUSION:

Yes, if

No, otherwise

Adjusting for a parent of Y is safer than a parent of X

WHICH SET TO ADJUST FOR

Should we adjust for {T}, {Z}, or {T, Z}?

Answer 1: (From bias-amplification considerations){T} is better than {T, Z} which is the same as {Z}

Answer 2: (From variance considerations){T} is better than {T, Z} which is better than {Z}

CONCLUSIONS

• The prevailing practice of adjusting for all covariates, especially those that are good predictors of X (the “treatment assignment,” Rubin, 2009) is totally misguided.

• The “outcome mechanism” is as important, and much safer, from both bias and variance viewpoints

• As X-rays are to the surgeon, graphs are for causation

REGRESSION VS. STRUCTURAL EQUATIONS(THE CONFUSION OF THE CENTURY)

Regression (claimless, nonfalsifiable): Y = ax + Y

Structural (empirical, falsifiable): Y = bx + uY

Claim: (regardless of distributions): E(Y | do(x)) = E(Y | do(x), do(z)) = bx

Q. When is b estimable by regression methods?A. Graphical criteria available

The mothers of all questions:Q. When would b equal a?A. When all back-door paths are blocked, (uY X)

TWO PARADIGMS FOR CAUSAL INFERENCE

Observed: P(X, Y, Z,...)

Conclusions needed: P(Yx=y), P(Xy=x | Z=z)...

How do we connect observables, X,Y,Z,…

to counterfactuals Yx, Xz, Zy,… ?N-R modelCounterfactuals areprimitives, new variables

Super-distribution

Structural modelCounterfactuals are derived quantities

Subscripts modify the model and distribution )()( yYPyYP xMx ,...,,,

,...),,...,,(*

zxZYZYX

constrain

“SUPER” DISTRIBUTIONIN N-R MODEL

Xy=0 0

inconsistency: x = 0 Yx=0 = Y Y = xY1 + (1-x) Y0

xyxzyx

ZYZYZYXP

...),...,...,,...,,(*

:Defines

ARE THE TWO PARADIGMS EQUIVALENT?

• Yes (Galles and Pearl, 1998; Halpern 1998)

• In the N-R paradigm, Yx is defined by consistency:

• In SCM, consistency is a theorem.

• Moreover, a theorem in one approach is a theorem in the other.

• Difference: Clarity of assumptions and their implications

01 )1( YxxYY

AXIOMS OF STRUCTURAL COUNTERFACTUALS

1. Definiteness

2. Uniqueness

3. Effectiveness

4. Composition (generalized consistency)

5. Reversibility

xuXtsXx y )( ..

')')((&))(( xxxuXxuX yy

xuX xw )(

)()()( uYuYxuX wwxw

yuYwuWyuY xxyxw )())((&))((

Yx(u)=y: Y would be y, had X been x (in state U = u)

(Galles, Pearl, Halpern, 1998):

FORMULATING ASSUMPTIONSTHREE LANGUAGES

},{),()(

),()()()(

XYZuYuY

uXuXuXuX

2. Counterfactuals:

1. English: Smoking (X), Cancer (Y), Tar (Z), Genotypes (U)

3. Structural:

1. Expressing scientific knowledge2. Recognizing the testable implications of one's

assumptions 3. Locating instrumental variables in a system of

equations4. Deciding if two models are equivalent or nested5. Deciding if two counterfactuals are independent

given another6. Algebraic derivations of identifiable estimands

COMPARISON BETWEEN THEN-R AND SCM LANGUAGES

GRAPHICAL – COUNTERFACTUALS SYMBIOSIS

Every causal graph expresses counterfactuals assumptions, e.g., X Y Z

consistent, and readable from the graph.

• Express assumption in graphs• Derive estimands by graphical or algebraic

methods

)()(, uYuY xzx

2. Missing arcs Y Z yx ZY

1. Missing arrows Y Z

EFFECT DECOMPOSITION(direct vs. indirect effects)

1. Why decompose effects?

2. What is the definition of direct and indirect effects?

3. What are the policy implications of direct and indirect effects?

4. When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?

WHY DECOMPOSE EFFECTS?

1. To understand how Nature works

2. To comply with legal requirements

3. To predict the effects of new type of interventions:

Signal routing, rather than variable fixing

LEGAL IMPLICATIONSOF DIRECT EFFECT

What is the direct effect of X on Y ?

(averaged over z)

))(),(())(),( 01 zdoxdoYEzdoxdoYECDE ||(

(Qualifications)

(Hiring)

(Gender)

Can data prove an employer guilty of hiring discrimination?

Adjust for Z? No! No!

FISHER’S GRAVE MISTAKE(after Rubin, 2005)

Compare treated and untreated lots of same density

No! No!

))(),(())(),( 01 zdoxdoYEzdoxdoYE ||(

(Plant density)

(Yield)

(Soil treatment)

What is the direct effect of treatment on yield?

(Latent factor)

Proposed solution (?): “Principal strata”

z = f (x, u)y = g (x, z, u)

NATURAL INTERPRETATION OFAVERAGE DIRECT EFFECTS

Natural Direct Effect of X on Y:The expected change in Y, when we change X from x0 to x1 and, for each u, we keep Z constant at whatever value it attained before the change.

In linear models, DE = Natural Direct Effect

][001 xZx YYE

);,( 10 YxxDE

Robins and Greenland (1992) – “Pure”

)( 01 xx

DEFINITION AND IDENTIFICATION OF NESTED COUNTERFACTUALS

Consider the quantity

Given M, P(u), Q is well defined

Given u, Zx*(u) is the solution for Z in Mx*, call it z

is the solution for Y in Mxz

Can Q be estimated from data?

Experimental: nest-free expressionNonexperimental: subscript-free expression

)]([ )(*uYEQ uxZxu

entalnonexperim

alexperiment

)()(*uY uxZx

z = f (x, u)y = g (x, z, u)

DEFINITION OFINDIRECT EFFECTS

Indirect Effect of X on Y:

The expected change in Y when we keep X constant, say at x0,

and let Z change to whatever value it would have attained had X

changed to x1.

In linear models, IE = TE - DE

][010 xZx YYE

);,( 10 YxxIE

No Controlled Indirect Effect

POLICY IMPLICATIONS OF INDIRECT EFFECTS

GENDER QUALIFICATION

HIRING

What is the indirect effect of X on Y?

The effect of Gender on Hiring if sex discriminationis eliminated.

IGNORE

Deactivating a link – a new type of intervention

1. The natural direct and indirect effects are identifiable in Markovian models (no confounding),

2. And are given by:

3. Applicable to linear and non-linear models, continuous and discrete variables, regardless of distributional form.

MEDIATION FORMULAS

))](|())(|())[,(|(

)).(|())],(|()),(|([

revIEDETE

xdozPxdozPzxdoYEIE

xdozPzxdoYEzxdoYEDE

IEDETE WHY

IEDETE

In linear systems

xzzmxy

mIEDETE

mmmTELinear + interaction

MEDIATION FORMULASIN UNCONFOUNDED MODELS

)|()|(

])|()|()[,|([

)|()],|(),|([

xYExYETE

xzPxzPzxYEIE

xzPzxYEzxYEDE

mediation to owed responses of Fraction

mediationby explained responses of Fraction

)(revIEDETE

DETEmmIE

mediation

disablingby prevented Effect

alone mediationby sustained Effect

IEDETE WHY

IErevIE )(

Disabling mediation

Disabling direct path

TE - DE

In linear systems

Is NOT equal to:

MEDIATION FORMULAFOR BINARY VARIABLES

)()1)((

000101

0011100010gghhIE

hgghggDE

XX ZZ YY E(Y|x,z)=gxz E(Z|x)=hx

nn11 00 00 00

nn22 00 00 11

nn33 00 11 00

nn44 00 11 11

nn55 11 00 00

nn66 11 00 11

nn77 11 11 00

nn88 11 11 11

43 hnnnn

87 hnnnn

RAMIFICATION OF THEMEDIATION FORMULA

• DE should be averaged over mediator levels,

IE should NOT be averaged over intervention levels.

• TE-DE need not equal IETE-DE = proportion for whom mediation is necessary

IE = proportion for whom mediation is sufficient

• TE-DE informs interventions on indirect pathways

IE informs intervention on direct pathways.

1. A Theory of causal transportabilityWhen can causal relations learned from experimentsbe transferred to a different environment in which no experiment can be conducted?

2. A Theory of statistical transportabilityWhen can statistical information learned in one domain be transferred to a different domain in whicha. only a subset of variables can be observed? Or,b. only a few samples are available?

3. A Theory of Selection BiasSelection bias occurs when samples used in learning are chosen differently than those used in testing. When can this bias be eliminated?

RECENT PROGRESS

TRANSPORTABILITY — WHEN CAN WE EXTRAPOLATE EXPERIMENTAL FINDINGS TO A

DIFFERENT ENVIRONMENT?

Experimental study in LAMeasured:

Needed:

)),(|(),,(

zxdoyPzyxP

? ))(|(* xdoyP

Observational study in NYCMeasured: ),,(* zyxP

),,(*),,( zyxPzyxP

X (Intervention)

Y (Outcome)

Z (Observation)

TRANSPORT FORMULAS DEPEND ON THE STORY

a) Z represents age

b) Z represents language skill

c) Z represents a bio-marker

zPzxdoyPxdoyP )(*)),(|())(|(*

))(|(* xdoyP

X Y(c)

))(|( xdoyP

xzPzxdoyP )|(*)),(|())(|(* xdoyP

TRANSPORTABILITY(Pearl and Bareinboim, 2011)

Definition 1 (Transportability)Given two environments, and *, characterized by graphs G and G*, a causal relation R is transportable from to * if

1. R() is estimable from the set I of interventional studies on , and

2. R(*) is identified from I, P*, G, and G*.

)(* tP

RESULT : ALGORITHM TO DETERMINEIF AN EFFECT IS TRANSPORTABLE

SINPUT: Causal Graph

OUTPUT:1. Transportable or not?2. Measurements to be taken in

the experimental study3. Measurements to be taken in

the target population4. A transport formula

S Factors creating differences

))(|(* xdoyP

),)(|( zxdoyP z

)|(* wzP t

),)(|( txdowP

X Y(f)

X Y(d)

WHICH MODEL LICENSES THE TRANSPORT OF THE CAUSAL EFFECT XY

X Y(e)

(c)X YZ

S External factors creating disparities

Yes YesNo

Yes NoYes

STATISTICAL TRANSPORTABILITY

Why should we transport statistical information?

Why not re-learn things from scratch ?1. Measurements are costly

Limit measurements to a subset V * of variables called “scope”.

2. Samples are costlyShare samples from diverse domains to illuminate their common components.

STATISTICAL TRANSPORTABILITY

Definition: (Statistical Transportability) A statistical relation R(P) is said to be transportable from to *

over V * if R(P*) is identified from P, P*(V *), and G* where P*(V *) is the marginal distribution of P* over a subset of variables V *.

S R=P* (y | x) is estimable without re-measuring Y

yzPxzPR )|()|(*

SIf few samples (N2) are available from * and many samples (N1) from, then estimating R = P*(y | x) by decomposition:

achieves much higher precision

xzPzxyPR )|(),|(*

No selection biasSelection bias activated by a virtual colliderSelection bias activated by both a virtual collider and real collider

THE ORIGIN OFSELECTION BIAS:

Can be eliminated by randomization or adjustment

CONTROLLING SELECTION BIAS BY ADJUSTMENT

Cannot be eliminated by randomization, requires adjustment for U2

Cannot be eliminated by randomization or adjustment

Cannot be eliminated by adjustment or by randomization

Adjustment for U2 gives

If all we have is P(u2 | S2 = 1), not P(u2), then only the U2-specific effect is recoverable

CONTROLLING BY ADJUSTMENT MAY REQUIRE EXTERNAL

INFORMATION U1 U2

222 )(),1,|())(|(u

uPuSxyPxdoyP

WHEN WOULD THE ODDSRATIO BE RECOVERABLE?

(a) OR is recoverable, despite the virtual collider at Y. whenever (X Y | Y,Z)G or (Y S | X , Z)G, giving:

OR (X,Y | S = 1) = OR(X,Y) (Cornfield 1951, Wittemore 1978; Geng 1992)

(b) the Z-specific OR is recoverable, but is meaningless. OR (X ,Y | Z) = OR (X,Y | Z, S = 1)

(c) the C-specific OR is recoverable, which is meaningful. OR (X ,Y | C) = OR (X,Y | W,C, S = 1)

(a) (b) (c)YX

)|()/(

/)|()|(

xyPxyP

Theorem (Bareinboim and Pearl, 2011)A necessary condition for the recoverability of OR(Y,X) is that every ancestor Ai of S that is also a descendant of X have a separating set that either:1. d-separates Ai from X given Y, or 2. d-separates Ai from Y given X.

Result: Polynomial algorithm to decide recoverability

2. GRAPHICAL CONDITION FORODDS-RATIO RECOVERABILITY

EXAMPLES OF ODDSRATIO RECOVERABILITY

(a) (b)

Recoverable Non-recoverable

)1,,,|,(),( 421 SWWWXYORXYOR (a)

{W1, W2, W4}{W4, W3, W1}

Separator

(a) (b)

Separator

(a) (b)

Separator

(a) (b)

Separator

(a) (b)

Separator

I TOLD YOU CAUSALITY IS SIMPLE

• Formal basis for causal and counterfactual inference

(complete)

• Unification of the graphical, potential-outcome and

structural equation approaches

• Friendly and formal solutions to

century-old problems and confusions.

• No other method can do better (theorem)

CONCLUSIONS

Thank you for agreeing with everything I said.

1 reasoning with causes and counterfactuals judea pearl ucla (judea/)

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