# Causal Inference As a Machine Learning Exercise REASONING WITH CAUSE AND EFFECT Judea Pearl Department of Computer Science UCLA 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 Robustness of Causal Claims TRADITIONAL STATISTICAL INFERENCE PARADIGM Data P Joint

Distribution Q(P) (Aspects of P) Inference e.g., Infer whether customers who bought product A would also buy product B. Q = P(B|A) THE CAUSAL INFERENCE PARADIGM Data M Data-generating Model Q(M) (Aspects of M)

Inference Some Q(M) cannot be inferred from P. e.g., Infer whether customers who bought product A would still buy A if we double the price. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations Statistics Data Probability joint distribution inferences from passive observations

FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations Statistics Probability inferences Data from passive observations Causal analysis deals with changes (dynamics) i.e. What remains invariant when P changes. joint distribution 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 Probability and statistics deal with static relations Statistics Probability inferences Data from passive observations Causal analysis deals with changes (dynamics) 1. Effects of Data interventions Causal 2. Causes of Model Causal effects assumptions 3. Explanations Experiments

joint distribution FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 1. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables 2. 3. 4. STATISTICAL Regression Association / Independence

Controlling for / Conditioning Odd and risk ratios Collapsibility FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 1. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility

2. No causes in no causes out (Cartwright, 1989) statistical assumptions + data causal conclusions causal assumptions } 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT) 1. Causal and statistical concepts do not mix. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables

STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility 2. No causes in no causes out (Cartwright, 1989) statistical assumptions + data causal conclusions causal assumptions } 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. Non-standard mathematics: a) Structural equation models (SEM) b) Counterfactuals (Neyman-Rubin) c) Causal Diagrams (Wright, 1920) WHAT'S IN A CAUSAL MODEL?

Oracle that assigns truth value to causal sentences: Action sentences: Counterfactuals: Explanation: B if we do A. B would be different if A were true. B occurred because of A. Optional: with what probability? FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION X Y Z INPUT

OUTPUT CAUSAL MODELS AND CAUSAL DIAGRAMS Definition: A causal model is a 3-tuple M = 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 AND CAUSAL DIAGRAMS Definition: A causal model is a 3-tuple M = 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 q b1 p d1i u1 p b2q d 2 w u2 U1 I W Q P U2 PAQ

CAUSAL MODELS AND MUTILATION Definition: A causal model is a 3-tuple M = 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 AND MUTILATION

Definition: A causal model is a 3-tuple M = 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) q b1 p d1i u1 p b2q d 2 w u2 U1 I W Q

P U2 CAUSAL MODELS AND MUTILATION Definition: A causal model is a 3-tuple M = 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) Mp U1 q b1 p d1i u1 p b2q d 2 w u2

I W U2 p p0 Q P P = p0 PROBABILISTIC CAUSAL MODELS Definition: A causal model is a 3-tuple M = 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. CAUSAL MODELS AND COUNTERFACTUALS Definition: Potential Response The 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 Response The 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) Joint probabilities of counterfactuals: P(Yx y, Z w z ) u:Yx (u ) y,Z w (u ) z P(u )

CAUSAL MODELS AND COUNTERFACTUALS Definition: Potential Response The 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) Joint probabilities of counterfactuals: P(Yx y, Z w z ) In particular: P(u ) u:Yx (u ) y,Z w (u ) z P ( y | do(x ) ) P(Yx y ) P(u )

u:Yx (u ) y P (Yx ' y '| x, y ) P(u | x, y ) u:Yx ' (u ) y ' 3-STEPS TO COMPUTING COUNTERFACTUALS S5. If the prisoner is dead, he would still be dead if A were not to have shot. DDA Abduction TRUE U (Court order) C (Captain) A

TRUE B (Riflemen) D (Prisoner) 3-STEPS TO COMPUTING COUNTERFACTUALS S5. If the prisoner is dead, he would still be dead if A were not to have shot. DDA Abduction TRUE Action U TRUE C

Prediction U TRUE C C FALSE FALSE A TRUE B D U A

B D A B D TRUE COMPUTING PROBABILITIES OF COUNTERFACTUALS 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) = ? Abduction P(u) P(u|D) Action U

P(u|D) C Prediction U P(u|D) C C FALSE FALSE A TRUE B D

U A B D A B D P(DA|D) CAUSAL INFERENCE MADE EASY (1985-2000) 1. Inference with Nonparametric Structural Equations made possible through Graphical Analysis. 2. Mathematical underpinning of counterfactuals through nonparametric structural equations 3. Graphical-Counterfactuals symbiosis

IDENTIFIABILITY Definition: 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 P(M1) = P(M2) Q(M1) = Q(M2) for all M1, M2, that satisfy A. IDENTIFIABILITY Definition: 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 P(M1) = P(M2) Q(M1) = Q(M2) for all M1, M2, that satisfy A. In other words, Q can be determined uniquely from the probability distribution P(v) of the

endogenous variables, V, and assumptions A. IDENTIFIABILITY Definition: 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 P(M1) = P(M2) Q(M1) = Q(M2) for all M1, M2, that satisfy A. In this talk: A: Assumptions encoded in the diagram Q1: P(y|do(x)) Causal Effect (= P(Yx=y)) Q2: P(Yx=y | x, y) Probability of necessity E (YxZ Direct ) Q3: Effect x' THE FUNDAMENTAL THEOREM

OF CAUSAL INFERENCE Causal Markov Theorem: Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as P(v1, v2,..., vn ) P(vi | pai ) i Where pai are the (values of) the parents of Vi in the causal diagram associated with M. THE FUNDAMENTAL THEOREM OF CAUSAL INFERENCE Causal Markov Theorem: Any distribution generated by Markovian structural model M (recursive, with independent disturbances) can be factorized as P(v1, v2,..., vn ) P(vi | pai ) i Where pai are the (values of) the parents of Vi in the causal diagram associated with M. 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

P(v1, v2,..., vn | do( x )) P(vi | pai ) | i|Vi X X x RAMIFICATIONS OF THE FUNDAMENTAL THEOREM Given P(x,y,z), should we ban smoking? U (unobserved) U (unobserved) X Smoking Z Tar in Lungs Pre-intervention

Y Cancer X=x Smoking Z Tar in Lungs Post-intervention Y Cancer RAMIFICATIONS OF THE FUNDAMENTAL THEOREM Given P(x,y,z), should we ban smoking?

U (unobserved) U (unobserved) X Z Smoking Tar in Lungs Y Cancer X=x Smoking Y Z

Tar in Lungs Cancer Pre-intervention Post-intervention P( x, y, z ) P(u )P( x | u )P( z | x )P( y | z, u ) P( y, z | do( x )) P(u ) P( z | x )P( y | z, u ) u u RAMIFICATIONS OF THE FUNDAMENTAL THEOREM

Given P(x,y,z), should we ban smoking? U (unobserved) U (unobserved) X Z Smoking Tar in Lungs Y Cancer X=x Smoking Y

Z Tar in Lungs Cancer Pre-intervention Post-intervention P( x, y, z ) P(u )P( x | u )P( z | x )P( y | z, u ) P( y, z | do( x )) P(u ) P( z | x )P( y | z, u ) u u To compute P(y,z|do(x)), we must eliminate u. (Graphical problem.) THE BACK-DOOR CRITERION

Graphical test of identification P(y | do(x)) is identifiable in G if there is a set Z of variables such that Z d-separates X from Y in Gx. G Z1 Z3 X Z1 Z2 Z4 Z6 Z Z3

Z5 Y Gx Z4 X Z6 Z2 Z5 Y THE BACK-DOOR CRITERION Graphical test of identification P(y | do(x)) is identifiable in G if there is a set Z of variables such that Z d-separates X from Y in Gx. G Z1 Z3

X Z1 Z2 Z4 Z6 Z Z3 Z5 Y Gx Z4 X Z6

Moreover, P(y | do(x)) = P(y | x,z) P(z) z (adjusting for Z) Z2 Z5 Y RULES OF CAUSAL CALCULUS Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w) if (Y Z|X,W ) G Rule 2: Action/observation exchange X P(y | do{x}, do{z}, w) = P(y | do{x},z,w) if (Y Z|X,W )G Rule 3: Ignoring actions

XZ P(y | do{x}, do{z}, w) = P(y | do{x}, w) if (Y Z|X,W )G X Z(W) DERIVATION IN CAUSAL CALCULUS Genotype (Unobserved) Smoking Tar Cancer P (c | do{s}) = t P (c | do{s}, t) P (t | do{s}) Probability Axioms = t P (c | do{s}, do{t}) P (t | do{s})

Rule 2 = t P (c | do{s}, do{t}) P (t | s) Rule 2 = t P (c | do{t}) P (t | s) Rule 3 = st P (c | do{t}, s) P (s | do{t}) P(t |s) Probability Axioms = st P (c | t, s) P (s | do{t}) P(t |s) Rule 2 = s t P (c | t, s) P (s) P(t |s) Rule 3 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 THAN NOT 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: PN ( x, y ) P(Yx' y' | x, y ) P(Yx' y' , X x,Y y ) P( X x,Y y ) 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 LEGAL RESPONSIBILITY? Deaths (y) Survivals (y) Experimental do(x) do(x) 16 14 984 986 1,000 1,000

Nonexperimental x x 2 28 998 972 1,000 1,000 Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival Plaintiff: Mr. A is special. 1. He actually died 2. He used the drug by choice Court to decide (given both data): Is it more probable than not that A would be alive but for the drug? PN P(Yx' y' | x, y ) 0.50

TYPICAL THEOREMS (Tian and Pearl, 2000) Bounds given combined nonexperimental and experimental data 0 1 P ( y ) P ( y ) P ( y' ) x' x' max PN min

P( x,y ) P ( x,y ) Identifiability under monotonicity (Combined data) P( y|x ) P( y|x' ) P( y|x' ) P( y x' ) PN P( y|x ) P( x,y ) corrected Excess-Risk-Ratio SOLUTION TO THE ATTRIBUTION PROBLEM (Cont)

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?

WHY DECOMPOSE EFFECTS? 1. Direct (or indirect) effect may be more transportable. 2. Indirect effects may be prevented or controlled. Pill Pregnancy + + Thrombosis 3. Direct (or indirect) effect may be forbidden Gender Qualification Hiring TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS IN LINEAR MODELS b

X a Z c z = bx + 1 y = ax + cz + 2 Y TE E (Y | do( x )) a + bc x DE E (Y | do( x ), do( z )) a x IE TE DE bc

Z - independen t SEMANTICS BECOMES NONTRIVIAL IN NONLINEAR MODELS (even when the model is completely specified) X Z z = f (x, 1) y = g (x, z, 2) Y TE E (Y | do( x )) x DE E (Y | do( x ), do( z )) x IE

???? Dependent on z? Void of operational meaning? THE OPERATIONAL MEANING OF DIRECT EFFECTS X Z z = f (x, 1) y = g (x, z, 2) Y 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. E[Yx1Z x Yx0 ] 0

In linear models, NDE = Controlled Direct Effect POLICY IMPLICATIONS (Who cares?) indirect What is the direct effect of X on Y? The effect of Gender on Hiring if sex discrimination is eliminated. GENDER X IGNORE Z QUALIFICATION f Y HIRING THE OPERATIONAL MEANING OF INDIRECT EFFECTS X Z z = f (x, 1)

y = g (x, z, 2) Y 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. E [Yx0 Z x Yx0 ] 1 In linear models, NIE = TE - DE LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION (FORMALIZING DISCRIMINATION) ``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 = female y = hire, y = not hire z = applicants qualifications NO DIRECT EFFECT Yx'Z Yx, x YxZ x' Yx' SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS Consider the quantity Q Eu [YxZ x * (u ) (u )]

Given M, P(u), Q is well defined Given u, Zx*(u) is the solution for Z in Mx*, call it z YxZ (u ) (u ) x* is the solution for Y in Mxz experiment al Can Q be estimated from nonexperim ental data? GRAPHICAL CONDITION FOR EXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS Theorem: If there exists a set W such that (Y Z | W )G XZ and W ND( X Z ) NDE ( x, x*;Y ) E (Yxz | w) E (Yx*z | w) P( Z x* z | w) P ( w) w, z Example: HOW THE PROOF GOES?

Proof: NDE x, x*;Y E Yx , Z x* E (Yx* ) If W Yxz Z x* | W for all z and x E (Yx, Z x* ) E (Yxz | Z x* z ,W w) w z P ( Z x* z | W w) P (W w) E (Yx, Z x* ) E (Yxz Y | W w) w z P ( Z x* z | W w) P (W w) Each factor is identifiable by experimentation.

GRAPHICAL CRITERION FOR COUNTERFACTUAL INDEPENDENCE Yxz Z x* | W for all z and x U3 U1 U 2 U3 U2 X Z Y U2 Z X U1 U1

Y U3 (Y Z | W )G XZ U1 U 2 U2 Z X Y U1 G XZ GRAPHICAL CONDITION FOR NONEXPERIMENTAL IDENTIFICATION OF AVERAGE NATURAL DIRECT EFFECTS NDE ( x, x*;Y )

E (Yxz | w) E (Yx*z | w) P ( Z x* z | w) P ( w) w, z Identification conditions 1. There exists a W such that (Y Z | W)GXZ 2. There exist additional covariates that render all counterfactual terms identifiable. IDENTIFICATION IN MARKOVIAN MODELS Corollary 3: The average natural direct effect in Markovian models is identifiable from nonexperimental data, and it is given by NDE ( x, x*;Y ) [ E (Y | x, z ) E (Y | x*, z )]P ( Z x* z ) z X Z

Y NDE ( x, x*;Y ) E (Y | x, z ) E ( y | x*, z ) P( z | x*) z RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS Theorem 5: The total, direct and indirect effects obey The following equality TE ( x, x*;Y ) NDE ( x, x*;Y ) NIE ( x*, x;Y ) In words, the total effect (on Y) associated with the transition from x* to x is equal to the difference between the direct effect associated with this transition and the indirect effect associated with the reverse transition, from x to x*. GENERAL PATH-SPECIFIC EFFECTS (Def.) X W

x* Z X W Y Z Y z* = Zx* (u) Form a new model, M g* , specific to active subgraph g fi* ( pai , u; g ) fi ( pai ( g ), pai*( g ), u ) Definition: g-specific effect E g ( x, x * ;Y )M TE ( x, x * ;Y ) M g*

Nonidentifiable even in Markovian models 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. Graphical conditions hold in Markovian models Useful in answering new type of policy questions involving mechanism blocking instead of variable fixing. THE OVERRIDING THEME 1. 2. Define Q(M) as a counterfactual expression Determine conditions for the reduction

Q( M ) Pexp ( M ) or Q( M ) P( M ) 3. If reduction is feasible, Q is inferable. Demonstrated on three types of queries: Q1: P(y|do(x)) Causal Effect (= P(Yx=y)) Q2: P(Yx = y | x, y) Probability of necessity E (YxZ Direct ) Q3: Effect x' 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

Actual Causation and Explanation Robustness of Causal Claims ROBUSTNESS: MOTIVATION Genetic Factors (unobserved) u x Smoking y Cancer In linear systems: y = on x cancer

+ The effect of smoking is, in general, is non-identifiable. non-identifiable (from observational studies). ROBUSTNESS: MOTIVATION Z Price of Cigarettes Genetic Factors (unobserved) u

y x Smoking Cancer Z Instrumental variable; cov(z,u) = 0 is identifiable R yz Rxz R yz Rxz ROBUSTNESS:

MOTIVATION Z Price of Cigarettes Genetic Factors (unobserved) u x Smoking y Cancer Problem with Instrumental Variables: The model may be wrong! R yz R yz

Rxz ROBUSTNESS: MOTIVATION Z1 Price of Cigarettes Z2 Peer Pressure Genetic Factors (unobserved) u

y x Smoking Cancer Solution: Invoke several instruments 1 R yz1 Rxz1 Surprise: 1 = 2 2 R yz2 Rxz2

model is likely correct ROBUSTNESS: MOTIVATION Z1 Price of Cigarettes Z2 Peer Pressure Genetic Factors (unobserved) u

x Smoking y Cancer Z3 Anti-smoking Legislation Zn Greater surprise: 1 = 2 = 3.= n = q Claim = q is highly likely to be correct ROBUSTNESS: MOTIVATION Genetic Factors (unobserved) u

x Smoking y Cancer s Symptom Symptoms do not act as instruments remains non-identifiable Why? Taking a noisy measurement (s) of an observed variable (y) cannot add new information ROBUSTNESS: MOTIVATION Genetic Factors (unobserved) Sn

u S2 x Smoking y Cancer S1 Symptom Adding many symptoms does not help. remains non-identifiable ROBUSTNESS: MOTIVATION

Given a parameter in a general graph x y Find if can evoke an equality surprise 1 = 2 = n associated with several independent estimands of Formulate: Surprise, over-identification, independence Robustness: The degree to which is robust to violations of model assumptions ROBUSTNESS: FORMULATION Bad attempt: if: f1, f2: Parameter is robust (over identifies)

f1() f 2 ( ) Two distinct functions if model induces constraint g () 0, then f () t1[ g ()] f () t2 [ g ()] ti [ g ()] are distinct. ROBUSTNESS: FORMULATION ex ey b x Ryx = b Rzx = bc Rzy = c

ez x = ex y = bx + ey z = cy + ez c y z (b) b R yx b Rzx / Rzy (c) c Rzy c Rzx / R yx

constraint: y z irrelvant to derivation of b Rzx R yx Rzy 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 b c x y Minimal assumption sets for c. x y c G1

z x y c z z x y G3 G2 Minimal assumption sets for b. c x

b y z z FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMS Definition A 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. e.g., Claim: (Total effect) TE(x,z) = q x y TE(x,z) = Rzx

z x x y y z TE(x,z) = Rzx Rzy x z FROM MINIMAL ASSUMPTION SETS TO MAXIMAL EDGE SUPERGRAPHS FROM PARAMETERS TO CLAIMS Definition A 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. e.g., Claim: (Total effect) TE(x,z) = q

x Nonparametric y TE ( x, z ) P ( z | x) z x x y z z y TE ( z , x) P ( y | x) P ( z | x' , y ) P ( x' ) y

x' 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. CONCLUSIONS Structural-model semantics enriched with logic + graphs leads to formal interpretation and practical assessments of wide variety of causal and counterfactual relationships.