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CAUSAL INFERENCE AS A MACHINE LEARNING
EXERCISE
Judea PearlComputer Science and Statistics
UCLAwww.cs.ucla.edu/~judea/
OUTLINE
• Learning: Statistical vs. Causal concepts
• Causal models and identifiability
• Learnability of three types of causal queries:
1. Effects of potential interventions,
2. Queries about attribution (responsibility)
3. Queries about direct and indirect effects
TRADITIONAL MACHINELEARNING PARADIGM
Data
Learning
Q(P)(Aspects of P)
PJoint
Distribution
e.g.,Learn whether customers who bought product Awould also buy product B.Q = P(B|A)
THE CAUSAL ANALYSISPARADIGM
Data
Learning
Q(M)(Aspects of M)
MData-generating
Model
Some Q(M) cannot be inferred from P.e.g.,Learn whether customers who bought product Awould still buy A if we double the price.• Data-mining vs. knowledge mining
THE SECRETS OFCAUSAL MODELS
Causal Model = Data-generating model satisfying:
1. Modularity (Symbol-mechanism correspondence)2. Uniqueness (Variable-mechanism correspondence)
THE SECRETS OFCAUSAL MODELS
Causal Model = Data-generating model satisfying:
Causal Model Joint DistributionPR = f (CP,PS, 1) P(PR, PS, CP, ME, QS)QS = g(ME, PR, 2); P(1, 2)
Q1: P(QS|PR=2) computable from P (and M)Q2: P(QS|do(PR=2) computable from M (not P)
f
Quantity Sold (QS)
g
Cost Proj.Prev. SaleOthers () Others ()
Marketing (MS)
PRICE (PR)
1. Modularity (Symbol-mechanism correspondence)2. Uniqueness (Variable-mechanism correspondence)
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
predictionsfrom 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 ratiosCollapsibilityPropensity score
1. Causal and statistical concepts do not mix.
2.
3.
4.
CAUSALSpurious correlationRandomizationConfounding / EffectInstrumentHolding constantExplanatory variables
STATISTICALRegressionAssociation / Independence“Controlling for” / ConditioningOdd and risk ratiosCollapsibilityPropensity score
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(iv)
U1 U2I W
Q P P = p0
0
222
111
pp
uwdqbp
uidpbq
Mp
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)
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:
),|(),|'(
)()()|(
')(:'
)(:
'
yxuPyxyYPN
uPyYPyP
yuYux
yuYux
x
x
In particular:
)(xdo
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
NON-PARAMETRICSTRUCTURAL MODELS
Given P(x,y,z), should we ban smoking?
x = u1,z = x + u2,y = z + u1 + u3.
Find: Find: P(y|do(x))
x = f1(u1),z = f2(x, u2),y = f3(z, u1, u3).
Linear Analysis Nonparametric Analysis
U
X Z Y
1
U2
Smoking Tar in Lungs
Cancer
U3
U
X Z Y
1
U2
Smoking Tar in Lungs
Cancer
U3
f1 f2f3
2f2
Given P(x,y,z), should we ban smoking?
x = u1,z = x + u2,y = z + u1 + u3.
Find: Find: P(y|do(x)) = P(Y=y) in new model
x = const.z = f2(x, u2),y = f3(z, u1, u3).
Linear Analysis Nonparametric Analysis
U
X = x Z Y
1
U
Smoking Tar in Lungs
Cancer
U3
U
X Z Y
1
U2
Smoking Tar in Lungs
Cancer
U3f3
LEARNING THE EFFECTS OF ACTIONS
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
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 WX,|ZY )( if
ZXGWXZY ),|( if
)(),|( if
WZXGWXZY
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)
A RECENTIDENTIFICATION RESULT
Theorem: [Tian and Pearl, 2001]The causal effect P(y|do(x)) is identifiable whenever the ancestral graph of Y contains no confounding path ( ) between X and any of its children.
Y
X
Z1
(b)
Z2
X
Z1
Y
(a)
Y
X
Z1
(c)
Z2
OUTLINE
• Learning: Statistical vs. Causal concepts
• Causal models and identifiability
• Learnability of three types of causal queries:
1. Distinguishing direct from indirect effects
2. Queries 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.”
2.
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 LEARNABLE 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
)()(
min)(
)()(max
x,yPy'P
PN x,yP
yPyP x'x'
10
)()()(
)()()(
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
• Learning: Statistical vs. Causal concepts
• Causal models and identifiability
• Learnability of three types of causal queries:
1. Effects of potential interventions,
2. Queries about attribution (responsibility)
3. Queries about direct and indirect effects
QUESTIONS ADDRESSED
• What is the semantics of direct and
indirect effects?
• Can we estimate them from data? Experimental data?
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
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
IDENTIFICATION INMARKOVIAN MODELS
X
Y
Z
z
xxZ
xzPzxYEzxYE
YEYEYxxDEx
*)|()]*,|(),|([
)()()*;,( **
]*)|()|()[*,|(
)()()*;,( **
z
xZx
xzPxzPzxYE
YEYEYxxIEx
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
CONCLUSIONS
• General theme:1. Define Q(M) as a counterfactual expression2. Determine conditions for the reduction
3. If reduction is feasible, Q is learnable.
• 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