REASONING WITH CAUSE AND EFFECT Judea Pearl Department of Computer Science UCLA

REASONING WITH CAUSE AND EFFECT

Judea PearlDepartment of Computer Science

UCLA

• 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

• Actual Causation and Explanation

• Falsifiability and Corroboration

OUTLINE

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

•

FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES

Datajoint

distribution

inferencesfrom passiveobservations

Probability and statistics deal with static relations

ProbabilityStatistics

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

FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES

Datajoint

distribution

inferencesfrom passiveobservations

Probability and statistics deal with static relations

ProbabilityStatistics

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.

CAUSALSpurious correlationRandomizationConfounding / EffectInstrumentHolding constantExplanatory variables

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

1. Causal and statistical concepts do not mix.

4.

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

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

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

statistical assumptions + datacausal assumptions causal conclusions }

CAUSALSpurious correlationRandomizationConfounding / EffectInstrumentHolding constantExplanatory variables

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

1. Causal and statistical concepts do not mix.

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.

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

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

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

U1 U2I W

Q P PAQ 222

111uwdqbp

uidpbq

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(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

CAUSAL MODELS ANDMUTILATION

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

U1 U2I W

Q P 222

111uwdqbp

uidpbq

(iv)

CAUSAL MODELS ANDMUTILATION

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(iv)

U1 U2I W

Q P P = p0

0

222

111

pp

uwdqbp

uidpbq

Mp

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(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”)

•

•

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:

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:

),|(),|'(

)()()|(

')(:'

)(:

'

yxuPyxyYP

uPyYPyP

yuYux

yuYux

x

x

In particular:

)(xdo

U

D

B

C

A

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

TRUE

Abduction

TRUE

(Court order)

(Captain)

(Riflemen)

(Prisoner)

U

D

B

C

A

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

TRUE U

D

B

C

A

FALSE

TRUE

Action

TRUE

U

D

B

C

A

FALSE

TRUE

PredictionAbduction

TRUE

U

D

B

C

A

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

Abduction

TRUE

Prediction

U

D

B

C

A

FALSE

P(u|D)

P(DA|D)

P(u)

Action

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)

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

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)

for all M1, M2, that satisfy A.

•

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)

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

Pre-intervention Post-intervention

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?

•

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?

Pre-intervention Post-interventionu

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

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

•

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?

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

• 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.

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

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.”

Answer:

),(),,'(

),|'(),(

'

'

yYxXPyYxXyYP

yxyYPyxPN

x

x

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.”

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

• 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

•

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)

NO DIRECT EFFECT

',' ' xxxx YYYYxZxZ

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

ACTUAL CAUSATION ANDACTUAL CAUSATION ANDTHE COUNTERFACTUAL TESTTHE COUNTERFACTUAL TEST

"We may define a cause to be an object followed byanother,..., where, if the first object had not been, thesecond never had existed."

Hume, Enquiry, 1748

Lewis (1973): "x CAUSED y " if x and y are true, and y is false in the closest non-x-world.

Structural interpretation:(i) X(u)=x(ii) Y(u)=y

(iii) Yx (u) y for x x

PROBLEM WITH THE COUNTERFACTUAL DEFINITION

Back-up to shoot iff Captain does not shoot at 12:00 noon

(Back-up)

(Prisoner)

(Captain)

Y

W

X

PROBLEM WITH THE COUNTERFACTUAL DEFINITION

Back-up to shoot iff Captain does not shoot at 12:00 noon

(Back-up)

(Prisoner)

(Captain)

Y

W

X

Scenario: Captain shot before noonPrisoner is dead

= 1

= 0

= 1

PROBLEM WITH THE COUNTERFACTUAL DEFINITION

Back-up to shoot iff Captain does not shoot at 12:00 noon

(Back-up)

(Prisoner)

(Captain)

Y

W

X

Scenario: Captain shot before noonPrisoner is dead

Q Is Captain’s shot the cause of death?A Yes, but the counterfactual test fails!

Intuition: Back-up might fall asleep – structural contingency

= 1

= 0

= 1

SELECTED STRUCTURAL CONTINGENCIES AND SUSTENANCE

x sustains y against W iff:(i) X(u) = x;(ii) Y(u) = y ;

(iii) Yxw(u) = y for all w; and(iv) Yx w (u) = y for some x x and some w

Y

W

X

= 1

= w = 0

= 1

Y

W

X

= 0

= 0

= w = 0

Definition: The explanatory power of a proposition X=x relative to an observed event Y=y is given by P(K{x,y}|x), the pre-discovery probability of the set of contexts K in which x is the actual cause of y.

FIRE

AND

Oxygen MatchOxygen

MatchKO,F = KM,F = K

EP(O) = P(K|O) << 1 EP(M) = P(K|M) 1

EXPLANATORY POWER(Halpern and Pearl, 2001)

CORRECTNESS and CORROBORATION

Data D corroborates structure S if S is (i) falsifiable and (ii) compatible with D.

Falsifiability: P*(S) P*

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

Constraints implied by S

P*P*(S)

D (Data)

e.g.,

w x y z

x

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

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

x yx ya

e.g., An un-corroborated structure, a is identifiable.

a = rYX

Intuitively, claim a = rYX is not corroborated because the assumptions that entail the claim are not falsifiable. i.e., no data falsifies the assumption rs = 0.

x yx ya

rs

a = rYX - rs

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

e.g., x y zx ya a

x y z

A corroborated structure can imply uncorroboratedclaims.

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

ax y ze.g., x y zx y

a

Some claims can be more corroborated than others.

a a b

b = rZY is corroborated because the assumptions needed for entailing this claim constrain the data by

YX

ZXZY r

rr

b

b b

FROM CORROBORATING MODELSTO CORROBORATING CLAIMS

Definition: An identifiable claim C is corroborated by data if the union of all minimal sets of assumptions sufficient for identifying C is corroborated by the data.

e.g., x y zx yaa aa

x y z

Some claims can be more corroborated than others.

abax y z

b

GRAPHICAL CRITERION FORCORROBORATED CLAIMS

Theorem: An identifiable claim C is corroborated by data if The intersection of all maximal supergraphs sufficientfor identifying C is corroborated by the data.

e.g., x yxa

x y z

b

GRAPHICAL CRITERION FORCORROBORATED CLAIMS

Theorem: An identifiable claim C is corroborated by data if The intersection of all maximal supergraphs sufficientfor identifying C is corroborated by the data.

e.g.,ba

x y zx yx

ax y z

abax y zx yx

bax y zx yx

bax y z

Intersection:x yx

Maximal supergraphs:

CONCLUSIONS

Structural-model semantics enriched

with logic + graphs leads to formal

interpretation and practical assessments

of wide variety of causal and counterfactual

relationships.