Causal Model Ying Nian Wu UCLA Department of Statistics July 13, 2007 IPAM Summer School
THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.
-
Upload
esmond-mcdaniel -
Category
Documents
-
view
230 -
download
2
Transcript of THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.
![Page 1: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/1.jpg)
THE MATHEMATICSOF CAUSAL MODELING
Judea PearlDepartment of Computer Science
UCLA
![Page 2: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/2.jpg)
• 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
• Robustness of Causal Claims
OUTLINE
![Page 3: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/3.jpg)
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)
![Page 4: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/4.jpg)
THE CAUSAL INFERENCEPARADIGM
Data
Inference
Q(M)(Aspects of M)
Data Generating
Model
Some Q(M) cannot be inferred from P.e.g.,Infer whether customers who bought product Awould still buy A if we were to double the price.
JointDistribution
![Page 5: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/5.jpg)
FROM STATISTICAL TO CAUSAL ANALYSIS:1. THE DIFFERENCES
Datajoint
distribution
inferencesfrom passiveobservations
Probability and statistics deal with static relations
ProbabilityStatistics
•
![Page 6: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/6.jpg)
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
![Page 7: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/7.jpg)
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
![Page 8: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/8.jpg)
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.
![Page 9: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/9.jpg)
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 }
![Page 10: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/10.jpg)
4. Non-standard mathematics:a) Structural equation models (Wright, 1920; Simon, 1960)
b) Counterfactuals (Neyman-Rubin (Yx), Lewis (x Y))
CAUSALSpurious correlationRandomizationConfounding / EffectInstrumentHolding constantExplanatory variables
STATISTICALRegressionAssociation / Independence“Controlling for” / ConditioningOdd and risk ratiosCollapsibility
1. Causal and statistical concepts do not mix.
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 }
![Page 11: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/11.jpg)
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?
![Page 12: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/12.jpg)
Z
YX
INPUT OUTPUT
FAMILIAR CAUSAL MODELORACLE FOR MANIPILATION
![Page 13: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/13.jpg)
WHY CAUSALITY NEEDS SPECIAL MATHEMATICS
Y = 2XX = 1
X = 1 Y = 2
Process information
Had X been 3, Y would be 6.If we raise X to 3, Y would be 6.Must “wipe out” X = 1.
Static information
SEM Equations are Non-algebraic:
![Page 14: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/14.jpg)
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•
![Page 15: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/15.jpg)
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
![Page 16: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/16.jpg)
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
![Page 17: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/17.jpg)
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)
![Page 18: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/18.jpg)
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
![Page 19: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/19.jpg)
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
![Page 20: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/20.jpg)
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) and U=u, is equal to y.
)(),()(,)(:
uPzZyYPzuZyuYu
wxwx
Joint probabilities of counterfactuals:•
![Page 21: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/21.jpg)
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)
![Page 22: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/22.jpg)
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
![Page 23: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/23.jpg)
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)
![Page 24: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/24.jpg)
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
![Page 25: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/25.jpg)
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)
![Page 26: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/26.jpg)
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.
•
![Page 27: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/27.jpg)
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:
![Page 28: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/28.jpg)
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
•
![Page 29: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/29.jpg)
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
![Page 30: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/30.jpg)
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 ),|()|()())(|,(
•
![Page 31: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/31.jpg)
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.)
![Page 32: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/32.jpg)
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
![Page 33: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/33.jpg)
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?
•
![Page 34: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/34.jpg)
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
![Page 35: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/35.jpg)
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
![Page 36: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/36.jpg)
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)
![Page 37: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/37.jpg)
RECENT RESULTS ON IDENTIFICATION
Theorem (Tian 2002):
We can identify P(v | do{x}) (x a singleton)
if and only if there is no child Z of X connected
to X by a bi-directed path.
X
Z Z
Z
k
1
![Page 38: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/38.jpg)
• Do-calculus is complete• A complete graphical criterion available
for identifying causal effects of any set on any set
• References: Shpitser and Pearl 2006 (AAAI, UAI)
RECENT RESULTS ON IDENTIFICATION (Cont.)
![Page 39: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/39.jpg)
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.
![Page 40: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/40.jpg)
DETERMINING THE CAUSES OF EFFECTS(The Attribution Problem)
• Your Honor! My client (Mr. A) died BECAUSE he used that drug.
•
![Page 41: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/41.jpg)
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
![Page 42: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/42.jpg)
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.”
•
![Page 43: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/43.jpg)
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
![Page 44: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/44.jpg)
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.
![Page 45: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/45.jpg)
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…
![Page 46: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/46.jpg)
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
![Page 47: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/47.jpg)
AXIOMS OF CAUSAL COUNTERFACTUALS
1. Definiteness
2. Uniqueness
3. Effectiveness
4. Composition
5. Reversibility
xuXtsXx y )( ..
')')((&))(( xxxuXxuX yy
xuX xw )(
)()()( uYuYwuW xxwx
yuYwuWyuY xxyxw )())((&)((
:)( yuYx Y would be y, had X been x (in state U = u)
![Page 48: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/48.jpg)
GRAPHICAL – COUNTERFACTUALS SYMBIOSIS
Every causal model implies constraints on counterfactuals
e.g.,
XZY
uYuY
yx
xzx
|
)()(,
consistent, and readable from the graph.
Every theorem in SEM is a theorem in N-R, and conversely.
![Page 49: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/49.jpg)
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.
![Page 50: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/50.jpg)
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
![Page 51: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/51.jpg)
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
![Page 52: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/52.jpg)
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
•
![Page 53: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/53.jpg)
QUESTIONS ADDRESSED
• What is the semantics of direct and
indirect effects?
• Can we estimate them from data? Experimental data?
![Page 54: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/54.jpg)
1. Direct (or indirect) effect may be more transportable.2. Indirect effects may be prevented or controlled.
3. Direct (or indirect) effect may be forbidden
WHY DECOMPOSEEFFECTS?
Pill
Thrombosis
Pregnancy
+
+
Gender
Hiring
Qualification
![Page 55: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/55.jpg)
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
![Page 56: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/56.jpg)
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
![Page 57: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/57.jpg)
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
![Page 58: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/58.jpg)
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
![Page 59: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/59.jpg)
Example:
Theorem: If there exists a set W such that
GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION
OF AVERAGE NATURAL DIRECT EFFECTS
zw
xzxxz wPwzZPwYEwYEYxxNDE,
** )()|()|()|()*;,(
)()|( ZXNDWWZYXZG and
![Page 60: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/60.jpg)
Corollary 3:The average natural direct effect in Markovian models is identifiable from nonexperimental data, and it is given by
IDENTIFICATION INMARKOVIAN MODELS
X Z
Y
z
xzPzxyEzxYEYxxNDE *)|()*,|(),|()*;,(
z x zZPzxYEzxYEYxxNDE )()]*,|(),|([)*;,( *
![Page 61: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/61.jpg)
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
![Page 62: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/62.jpg)
CONCLUSIONS
Structural-model semantics enriched
with logic + graphs leads to formal
interpretation and practical assessments
of wide variety of causal and counterfactual
relationships.
![Page 63: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/63.jpg)
• 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
• Robustness of Causal Claims
OUTLINE
![Page 64: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/64.jpg)
ROBUSTNESS:MOTIVATION
Smoking
x y
Genetic Factors (unobserved)
Cancer
u
In linear systems: y = x + u cov (x,u) = 0 is identifiable. = Ryx
![Page 65: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/65.jpg)
ROBUSTNESS:MOTIVATION
The claim Ryx is sensitive to the assumption cov (x,u) = 0.
Smoking
x
Genetic Factors (unobserved)
Cancer
is non-identifiable if cov (x,u) ≠ 0.
y
u
![Page 66: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/66.jpg)
ROBUSTNESS:MOTIVATION
Z – Instrumental variable; cov(z,u) = 0
Smoking
y
Genetic Factors (unobserved)
Cancer
u
x
ZPrice ofCigarettes
xz
yz
xz
yzR
R
R
R
is identifiable, even if cov (x,u) ≠ 0
![Page 67: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/67.jpg)
ROBUSTNESS:MOTIVATION
Smoking
y
Genetic Factors (unobserved)
Cancer
u
x
ZPrice ofCigarettes
10
10
Suppose
xz
yzyx R
RR
Claim “ = Ryx” is likely to be true
![Page 68: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/68.jpg)
Smoking
ROBUSTNESS:MOTIVATION
Z1
Price ofCigarettes
Invoking several instruments
If =1 = 2, claim “ = 0” is more likely correct
2
22
1
10
xz
yz
xz
yzyx R
R
R
RR 1
x y
Genetic Factors (unobserved)
Cancer
u
PeerPressure
Z2
![Page 69: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/69.jpg)
ROBUSTNESS:MOTIVATION
Z1
Price ofCigarettes
x y
Genetic Factors (unobserved)
Cancer
u
PeerPressure
Z2
Smoking
Greater surprise: 1 = 2 = 3….= n = qClaim = q is highly likely to be correct
Z3
Zn
Anti-smoking Legislation
![Page 70: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/70.jpg)
ROBUSTNESS:MOTIVATION
Assume we have several independent estimands of , and
x y
Given a parameter in a general graph
Find the degree to which is robust to violations of model assumptions
1 = 2 = …n
![Page 71: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/71.jpg)
ROBUSTNESS:ATTEMPTED FORMULATION
Bad attempt: Parameter is robust (over-identified)
f1, f2: Two distinct functions
)()( 21 ff
distinct. are
then constraint induces model if
)]([
)]([)()]([)(
,0)(
21
gt
gtfgtf
g
i
if:
![Page 72: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/72.jpg)
ROBUSTNESS:MOTIVATION
x y
Genetic Factors (unobserved)
Cancer
u
Smoking
Is robust if 0 = 1?
sxsysy
sx
yx
RRRR
R
,1
0
s
Symptom
![Page 73: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/73.jpg)
ROBUSTNESS:MOTIVATION
x y
Genetic Factors (unobserved)
Cancer
u
Smoking
Symptoms do not act as instruments
remains non-identifiable if cov (x,u) ≠ 0
s
Symptom
Why? Taking a noisy measurement (s) of an observed variable (y) cannot add new information
![Page 74: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/74.jpg)
ROBUSTNESS:MOTIVATION
x
Genetic Factors (unobserved)
Cancer
u
Smoking
Adding many symptoms does not help.
remains non-identifiable
ySymptom
S1
S2
Sn
![Page 75: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/75.jpg)
INDEPENDENT:BASED ON DISTINCT SETS OF ASSUMPTION
u
z yx
u
zyx
EstimandEstimand AssumptiomsAssumptioms
xz
yz
yx
R
R
R
1
0
others
0),cov( ux
EstimandEstimand AssumptiomsAssumptioms
zy
zx
yx
RR
R
1
0
others
0),cov(
0),cov(
ux
ux
![Page 76: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/76.jpg)
RELEVANCE:FORMULATION
Definition 8 Let A be an assumption embodied in model M, and p a parameter in M. A is said to be relevant to p if and only if there exists a set of assumptions S in M such that S and A sustain the identification of p but S alone does not sustain such identification.
Theorem 2 An assumption A is relevant to p if and only if A is a member of a minimal set of assumptions sufficient for identifying p.
![Page 77: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/77.jpg)
ROBUSTNESS:FORMULATION
Definition 5 (Degree of over-identification)A parameter p (of model M) is identified to degree k (read: k-identified) if there are k minimal sets of assumptions each yielding a distinct estimand of p.
![Page 78: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/78.jpg)
ROBUSTNESS:FORMULATION
x y
b
z
c
Minimal assumption sets for c.
xy z
c xy z
c
G3G2
xy z
c
G1
Minimal assumption sets for b. xy
bz
![Page 79: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/79.jpg)
FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS
FROM PARAMETERS TO CLAIMS
DefinitionA claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand.
TE(x,z) = Rzx TE(x,z) = Rzx Rzy ·x
xy zx
y z
e.g., Claim: (Total effect) TE(x,z) = q x y z
![Page 80: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/80.jpg)
FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS
FROM PARAMETERS TO CLAIMS
DefinitionA claim C is identified to degree k in model M (graph G), if there are k edge supergraphs of G that permit the identification of C, each yielding a distinct estimand.
xy zx
y z
e.g., Claim: (Total effect) TE(x,z) = q x y z
Nonparametric y x
xPyxzPxyPxzTExzPzxTE'
)'(),'|()|(),()|(),(
![Page 81: THE MATHEMATICS OF CAUSAL MODELING Judea Pearl Department of Computer Science UCLA.](https://reader036.fdocuments.net/reader036/viewer/2022062407/56649d025503460f949d5f55/html5/thumbnails/81.jpg)
SUMMARY OF ROBUSTNESS RESULTS
1. Formal definition to ROBUSTNESS of causal claims:“A claim is robust when it is insensitive to
violations of some of the model assumptions relevant to substantiating that claim.”
2. Graphical criteria and algorithms for computing the degree of robustness of a given causal claim.