CAUSAL MODELING AND THE LOGIC OF SCIENCE

CAUSAL MODELINGAND THE

LOGIC OF SCIENCE

Judea PearlComputer Science and Statistics

UCLAwww.cs.ucla.edu/~judea/

OVERVIEWScope and Language in Scientific Theories

1. Statistical models (observtions, PL)

2. Causal models2.1 Stochastic causal model (interventions, PL + modality)2.2 Functional causal models (counterfactuals, PL + subjunctives)

3. General equational models (explicit interventions, PL)

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

4. General Scientific theories (objects-properties, FOL-SOL ...)

OUTLINE

• Modeling: Statistical vs. Causal

• Causal models and identifiability

• Inference to three types of claims:

1. Effects of potential interventions,

2. Claims about attribution (responsibility)

3. Claims about direct and indirect effects

• Falsifiability and Corroboration

TRADITIONAL STATISTICALINFERENCE PARADIGM

Data

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)

THE CAUSAL INFERENCEPARADIGM

Data

Inference

Q(M)(Aspects of M)

MData-generating

Model

Some Q(M) cannot be inferred from P.e.g.,Infer whether customers who bought product Awould still buy A if we double the price.

FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES

Datajoint

distribution

inferencesfrom passiveobservations

Probability and statistics deal with static relations

ProbabilityStatistics

•


Datajoint

distribution




Causal analysis deals with changes (dynamics)i.e. What remains invariant when P changes.

• P does not tell us how it ought to change

e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation


Datajoint

distribution




CausalModel

Data

Causalassumptions

1. Effects of interventions

2. Causes of effects

3. Explanations

Causal analysis deals with changes (dynamics)

Experiments

FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES (CONT)

CAUSALSpurious correlationRandomizationConfounding / EffectInstrumentHolding constantExplanatory variables

STATISTICALRegressionAssociation / Independence“Controlling for” / ConditioningOdd and risk ratiosCollapsibility

1. Causal and statistical concepts do not mix.

2.

3.

4.




4.

3. Causal assumptions cannot be expressed in the mathematical language of standard statistics.


2. No causes in – no causes out (Cartwright, 1989)

statistical assumptions + datacausal assumptions causal conclusions }




4. Non-standard mathematics:a) Structural equation models (SEM)b) Counterfactuals (Neyman-Rubin)c) Causal Diagrams (Wright, 1920)

3. Causal assumptions cannot be expressed in the mathematical language of standard statistics.


2. No causes in – no causes out (Cartwright, 1989)

statistical assumptions + datacausal assumptions causal conclusions }

WHAT'S IN A CAUSAL MODEL?

Oracle that assigns truth value to causalsentences:

Action sentences: B if we do A.

Counterfactuals: B would be different ifA were true.

Explanation: B occurred because of A.

Optional: with what probability?

Z

YX

INPUT OUTPUT

FAMILIAR CAUSAL MODELORACLE FOR MANIPILATION

CAUSAL MODELS ANDCAUSAL DIAGRAMS

Definition: A causal model is a 3-tupleM = V,U,F

with a mutilation operator do(x): M Mx where:

(i) V = {V1…,Vn} endogenous variables,(ii) U = {U1,…,Um} background variables(iii) F = set of n functions, fi : V \ Vi U Vi

vi = fi(pai,ui) PAi V \ Vi Ui U•

CAUSAL MODELS ANDCAUSAL DIAGRAMS




vi = fi(pai,ui) PAi V \ Vi Ui U

U1 U2I W

Q P PAQ 222

111uwdqbp

uidpbq




vi = fi(pai,ui) PAi V \ Vi Ui U(iv) Mx= U,V,Fx, X V, x X

where Fx = {fi: Vi X } {X = x}(Replace all functions fi corresponding to X with the constant

functions X=x)•

CAUSAL MODELS ANDMUTILATION





vi = fi(pai,ui) PAi V \ Vi Ui U

U1 U2I W

Q P 222

111uwdqbp

uidpbq

(iv)





vi = fi(pai,ui) PAi V \ Vi Ui U(iv)

U1 U2I W

Q P P = p0

0

222

111

pp

uwdqbp

uidpbq

Mp




vi = fi(pai,ui) PAi V \ Vi Ui U(iv) Mx= U,V,Fx, X V, x X

where Fx = {fi: Vi X } {X = x}(Replace all functions fi corresponding to X with the constant

functions X=x)

Definition (Probabilistic Causal Model): M, P(u)P(u) is a probability assignment to the variables in U.

PROBABILISTIC CAUSAL MODELS

CAUSAL MODELS AND COUNTERFACTUALS

Definition: Potential ResponseThe sentence: “Y would be y (in unit u), had X been x,”denoted Yx(u) = y, is the solution for Y in a mutilated model Mx, with the equations for X replaced by X = x. (“unit-based potential outcome”)

•

•



•

)(),()(,)(:

uPzZyYPzuZyuYu

wxwx

Joint probabilities of counterfactuals:



)(),()(,)(:

uPzZyYPzuZyuYu

wxwx

Joint probabilities of counterfactuals:

),|(),|'(

)()()|(

')(:'

)(:

'

yxuPyxyYPN

uPyYPyP

yuYux

yuYux

x

x

In particular:

)(xdo

U

D

B

C

A

Abduction Action Prediction

S5. If the prisoner is dead, he would still be dead if A were not to have shot. DDA

3-STEPS TO COMPUTING3-STEPS TO COMPUTINGCOUNTERFACTUALSCOUNTERFACTUALS

U

D

B

C

A

FALSE

TRUE

TRUE

U

D

B

C

A

FALSE

TRUETRUE

TRUE

U

D

B

C

A

Abduction

P(S5). The prisoner is dead. How likely is it that he would be dead if A were not to have shot. P(DA|D) = ?

COMPUTING PROBABILITIESCOMPUTING PROBABILITIESOF COUNTERFACTUALSOF COUNTERFACTUALS

Action

TRUE

Prediction

U

D

B

C

A

FALSE

P(u|D)

P(DA|D)

P(u) U

D

B

C

A

FALSE

P(u|D)

P(u|D)

CAUSAL INFERENCEMADE EASY (1985-2000)

1. Inference with Nonparametric Structural Equations made possible through Graphical Analysis.

2. Mathematical underpinning of counterfactualsthrough nonparametric structural equations

3. Graphical-Counterfactuals symbiosis

IDENTIFIABILITYIDENTIFIABILITYDefinition:Let Q(M) be any quantity defined on a causal model M, and let A be a set of assumption.

Q is identifiable relative to A iff

•

for all M1, M2, that satisfy A.

•

P(M1) = P(M2) Q(M1) = Q(M2)



In other words, Q can be determined uniquelyfrom the probability distribution P(v) of the endogenous variables, V, and assumptions A.

P(M1) = P(M2) Q(M1) = Q(M2)


•




P(M1) = P(M2) Q(M1) = Q(M2)

A: Assumptions encoded in the diagramQ1: P(y|do(x)) Causal Effect (= P(Yx=y))Q2: P(Yx=y | x, y) Probability of necessityQ3: Direct Effect)(

'xZxYE

In this talk:

THE FUNDAMENTAL THEOREMOF CAUSAL INFERENCE

Causal Markov Theorem:Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as

Where pai are the (values of) the parents of Vi in the causal diagram associated with M.

)|(),...,,( iii

n pavPvvvP 21

•

THE FUNDAMENTAL THEOREMOF CAUSAL INFERENCE

Causal Markov Theorem:Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as

Where pai are the (values of) the parents of Vi in the causal diagram associated with M.

)|(),...,,( iii

n pavPvvvP 21

xXXViiin

i

pavPxdovvvP

|)|( ))(|,...,,(|

21

Corollary: (Truncated factorization, Manipulation Theorem)The distribution generated by an intervention do(X=x)(in a Markovian model M) is given by the truncated factorization

RAMIFICATIONS OF THE FUNDAMENTAL THEOREM

U (unobserved)

X = x Z YSmoking Tar in

LungsCancer

U (unobserved)

X Z YSmoking Tar in

LungsCancer

Given P(x,y,z), should we ban smoking?

• •

•


U (unobserved)


LungsCancer

U (unobserved)

X Z YSmoking Tar in

LungsCancer


Pre-intervention Post-interventionu

uzyPxzPuxPuPzyxP ),|()|()|()(),,( u

uzyPxzPuPxdozyP ),|()|()())(|,(

•


U (unobserved)


LungsCancer

U (unobserved)

X Z YSmoking Tar in

LungsCancer


Pre-intervention Post-interventionu

uzyPxzPuxPuPzyxP ),|()|()|()(),,( u

uzyPxzPuPxdozyP ),|()|()())(|,(

To compute P(y,z|do(x)), we must eliminate u. (graphical problem).

THE BACK-DOOR CRITERIONGraphical test of identificationP(y | do(x)) is identifiable in G if there is a set Z ofvariables such that Z d-separates X from Y in Gx.

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z

•

Gx G

THE BACK-DOOR CRITERIONGraphical test of identificationP(y | do(x)) is identifiable in G if there is a set Z ofvariables such that Z d-separates X from Y in Gx.

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z6

Z3

Z2

Z5

Z1

X Y

Z4

Z

Moreover, P(y | do(x)) = P(y | x,z) P(z)(“adjusting” for Z) z

Gx G

RULES OF CAUSAL CALCULUSRULES OF CAUSAL CALCULUS

Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w)

Rule 2: Action/observation exchange P(y | do{x}, do{z}, w) = P(y | do{x},z,w)

Rule 3: Ignoring actions P(y | do{x}, do{z}, w) = P(y | do{x}, w)

XG Z|X,WY )( if

Z(W)XGZ|X,WY )( if

ZXGZ|X,WY )( if

DERIVATION IN CAUSAL CALCULUSDERIVATION IN CAUSAL CALCULUS

Smoking Tar Cancer

P (c | do{s}) = t P (c | do{s}, t) P (t | do{s})

= st P (c | do{t}, s) P (s | do{t}) P(t |s)

= t P (c | do{s}, do{t}) P (t | do{s})

= t P (c | do{s}, do{t}) P (t | s)

= t P (c | do{t}) P (t | s)

= s t P (c | t, s) P (s) P(t |s)

= st P (c | t, s) P (s | do{t}) P(t |s)

Probability Axioms

Probability Axioms

Rule 2

Rule 2

Rule 3

Rule 3

Rule 2

Genotype (Unobserved)

OUTLINE






3.

DETERMINING THE CAUSES OF EFFECTS(The Attribution Problem)

• Your Honor! My client (Mr. A) died BECAUSE he used that drug.

•

DETERMINING THE CAUSES OF EFFECTS(The Attribution Problem)

• Your Honor! My client (Mr. A) died BECAUSE he used that drug.

• Court to decide if it is MORE PROBABLE THANNOT that A would be alive BUT FOR the drug!

P(? | A is dead, took the drug) > 0.50

THE PROBLEM

Theoretical Problems:

1. What is the meaning of PN(x,y):“Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”

•

THE PROBLEM



Answer:

),(),,'(

),|'(),(

'

'

yYxXPyYxXyYP

yxyYPyxPN

x

x

THE PROBLEM



2. Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.

WHAT IS INFERABLE FROM EXPERIMENTS?

Simple Experiment:Q = P(Yx= y | z)Z nondescendants of X.

Compound Experiment:Q = P(YX(z) = y | z)

Multi-Stage Experiment:etc…

CAN FREQUENCY DATA DECIDE CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY?LEGAL RESPONSIBILITY?

• Nonexperimental data: drug usage predicts longer life• Experimental data: drug has negligible effect on survival

Experimental Nonexperimental do(x) do(x) x x

Deaths (y) 16 14 2 28Survivals (y) 984 986 998 972

1,000 1,000 1,000 1,000

1. He actually died2. He used the drug by choice

500.),|'( ' yxyYPPN x

• Court to decide (given both data): Is it more probable than not that A would be alive but for the drug?

• Plaintiff: Mr. A is special.

TYPICAL THEOREMS(Tian and Pearl, 2000)

• Bounds given combined nonexperimental and experimental data

)()(

1

min)(

)()(

0

maxx,yPy'P

PN x,yP

yPyP x'x'

)()()(

)()()(

x,yPyPy|x'P

y|xPy|x'Py|xP

PN x'

• Identifiability under monotonicity (Combined data)

corrected Excess-Risk-Ratio

SOLUTION TO THE ATTRIBUTION SOLUTION TO THE ATTRIBUTION PROBLEM (Cont)PROBLEM (Cont)

• WITH PROBABILITY ONE P(yx | x,y) =1

• From population data to individual case• Combined data tell more that each study alone

OUTLINE






3. Claims about direct and indirect effects

•

QUESTIONS ADDRESSED

• What is the semantics of direct and

indirect effects?

• Can we estimate them from data? Experimental data?

tindependen- ))(),(|(

))(|(

DETEIE

ZzdoxdoYEx

DE

xdoYEx

TE

TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS

IN LINEAR MODELS

X Z

Y

ca

b z = bx + 1

y = ax + cz + 2

a + bc

bc

a

z = f (x, 1)y = g (x, z, 2)

????

))(),(|(

))(|(

IE

zdoxdoYEx

DE

xdoYEx

TE

X Z

Y

SEMANTICS BECOMES NONTRIVIALIN NONLINEAR MODELS

(even when the model is completely specified)

Dependent on z?

Void of operational meaning?

z = f (x, 1)y = g (x, z, 2)

X Z

Y

THE OPERATIONAL MEANING OFDIRECT EFFECTS

“Natural” Direct Effect of X on Y:The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change.

In linear models, NDE = Controlled Direct Effect

][001 xZx YYE

x

POLICY IMPLICATIONS(Who cares?)

f

GENDER QUALIFICATION

HIRING

What is the direct effect of X on Y?

The effect of Gender on Hiring if sex discrimination is eliminated.

indirect

X Z

Y

IGNORE

z = f (x, 1)y = g (x, z, 2)

X Z

Y

THE OPERATIONAL MEANING OFINDIRECT EFFECTS

“Natural” 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 under a unit change in X.

In linear models, NIE = TE - DE

][010 xZx YYE

x

``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’

[Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]

x = male, x = femaley = hire, y = not hirez = applicant’s qualifications

LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION

(FORMALIZING DISCRIMINATION)

YxZx = Yx, YxZx

= Yx

NO DIRECT EFFECT

SEMANTICS 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?

)]([ )(*uYEQ uxZxu

entalnonexperim

alexperiment

)()(*uY uxZx

ANSWERS TO QUESTIONS

• Graphical conditions for estimability from experimental / nonexperimental data.

•

• Graphical conditions hold in Markovian models

ANSWERS TO QUESTIONS

• Graphical conditions for estimability from experimental / nonexperimental data.

• Useful in answering new type of policy questions involving mechanism blocking instead of variable fixing.

• Graphical conditions hold in Markovian models

THE OVERRIDING THEME

1. Define Q(M) as a counterfactual expression2. Determine conditions for the reduction

3. If reduction is feasible, Q is inferable.

• Demonstrated on three types of queries:

)()()()( exp MPMQMPMQ or

Q1: P(y|do(x)) Causal Effect (= P(Yx=y))Q2: P(Yx = y | x, y) Probability of necessityQ3: Direct Effect)(

'xZxYE

FALSIFIABILITY and CORROBORATION

Data D corroborates model M if M is (i) falsifiable and (ii) compatible with D.

Falsifiability: P*(M) P*

Types of constraints:1. conditional independencies2. inequalities (for restricted domains)3. functional

Constraints implied by M

P*P*(M)

D (Data)

e.g.,

w x y z

x

zyfyxwzPwxP ),(),,|()|(

OTHER TESTABLE CLAIMSChanges under interventions

For all causal models:zyxxyYPyYP zxz ,, )|()(

For all semi-Markovian models:

),,(),(

),()(

zyxPzZyYP

yYxXPyYP

xx

zzxz

For Markovian models (and ):VZYX

For a given Markovian model:)|()( \ YYv payPyYP

)()(),( yYPxXPxXyYP xzyzzz

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

A corroborated model can imply identifiable yetuncorroborated claims.

Some claims can be more corroborated than others.

Definition: An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.

Graphical criterion: minimal submodel = maximal supergraph

e.g.,a

x y z x y zx y za b

A corroborated model can imply identifiable yetuncorroborated claims.

Some claims can be more corroborated than others.

Definition: An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.

Graphical criterion: minimal submodel = maximal supergraph

e.g.,a

x y zx y za b

x y z

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

OVERVIEWScope and Language in Scientific Theories

1. Statistical models (observtions, PL)

2. Causal models2.1 Stochastic causal model (interventions, PL + modality)2.2 Functional causal models (counterfactuals, PL + subjunctives)

3. General equational models (explicit interventions, PL)

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

4. General Scientific theories (objects-properties, FOL-SOL ...)

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CAUSAL MODELING AND THE LOGIC OF SCIENCE