the mathematics of causal modeling judea pearl department of computer science ucla
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THE MATHEMATICSOF CAUSAL MODELING
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
• Robustness of Causal Claims
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)
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
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 }
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 }
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
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:
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: 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:•
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
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
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?
•
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)
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
• 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.)
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…
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
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)
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.
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?
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
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
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
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
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
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 )()]*,|(),|([)*;,( *
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
CONCLUSIONS
Structural-model semantics enriched
with logic + graphs leads to formal
interpretation and practical assessments
of wide variety of causal and counterfactual
relationships.
• 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
ROBUSTNESS:MOTIVATION
Smoking
x y
Genetic Factors (unobserved)
Cancer
u
In linear systems: y = x + u cov (x,u) = 0 is identifiable. = Ryx
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
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
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
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
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
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
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:
ROBUSTNESS:MOTIVATION
x y
Genetic Factors (unobserved)
Cancer
u
Smoking
Is robust if 0 = 1?
sxsysy
sx
yx
RRRR
R
,1
0
s
Symptom
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
ROBUSTNESS:MOTIVATION
x
Genetic Factors (unobserved)
Cancer
u
Smoking
Adding many symptoms does not help.
remains non-identifiable
ySymptom
S1
S2
Sn
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
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.
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.
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
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
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'
)'(),'|()|(),()|(),(
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.