OF CAUSAL INFERENCE THE MATHEMATICS IN STATISTICS. Department of Computer Science. Judea Pearl UCLA
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1 THE MATHEMATICS OF CAUSAL INFERENCE IN STATISTICS Judea earl Department of Computer Science UCLA
2 OUTLINE Statistical vs. Causal Modeling: distinction and mental barriers N-R vs. structural model: strengths and weaknesses Formal semantics for counterfactuals: definition, axioms, graphical representations Graphs and Algebra: Symbiosis translation and accomplishments
3 TRADITIONAL STATISTICAL INFERENCE ARADIGM Data Joint Distribution Q( (Aspects of Inference e.g., Infer whether customers who bought product A would also buy product B. Q (B A
4 FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES robability and statistics deal with static relations Data Joint Distribution change Joint Distribution Q( (Aspects of Inference What happens when changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price.
5 FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES What remains invariant when changes say, to satisfy (price21 Data Joint Distribution change Joint Distribution Q( (Aspects of Inference Note: (v (v price 2 does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation
6 FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT 1. Causal and statistical concepts do not mix. 2. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility 3. 4.
7 FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES (CONT 1. Causal and statistical concepts do not mix. 4. CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables 2. No causes in no causes out (Cartwright, 1989 STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility statistical assumptions + data } causal assumptions causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics.
8 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. No causes in no causes out (Cartwright, 1989 STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility statistical assumptions + data } causal assumptions causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. Non-standard mathematics: a Structural equation models (Wright, 192; Simon, 196 b Counterfactuals (Neyman-Rubin (Y x, Lewis (x Y
9 TWO ARADIGMS FOR CAUSAL INFERENCE Observed: (X, Y, Z,... Conclusions needed: (Y x y, (X y x Zz... How do we connect observables, X,Y,Z, to counterfactuals Y x, X z, Z y,? N-R model Counterfactuals are primitives, new variables Super-distribution *(X, Y,, Y x, X z, X, Y, Z constrain Y x, Z y, Structural model Counterfactuals are derived quantities Subscripts modify the distribution (Y x y x (Yy
10 SUER DISTRIBUTION IN N-R MODEL X Y Z Y x Y x1 X z X z1 X y inconsistency: x Y x Y Y xy 1 + (1-x Y Defines : *( X, Y *( Y Y x x X, Z y Z y,... Y x, Z y Z, X z... Y xz, Z xy, U u 1 u 2 u 3 u 4
11 TYICAL INFERENCE IN N-R MODEL Find *(Y x y given covariate Z, Assume ignorability: Y x X Z Assume consistency: Xx Y x Y *( Y x y z z z z *( *( *( ( y Y Y Y x x x, z y y y x ( z x, z, z z ( z ( ( z z roblems: 1 2? Try it: X Y Z Y x X Z judgmental & opaque Is consistency the only connection between X, Y and Y x?
12 THE STRUCTURAL MODEL ARADIGM Data Joint Distribution Data Generating Model Q(M (Aspects of M Inference M Oracle for computing answers to Q s. e.g., Infer whether customers who bought product A would still buy A if we were to double the price.
13 FAMILIAR CAUSAL MODEL ORACLE FOR MANIILATION X Y Z INUT OUTUT
14 STRUCTURAL CAUSAL MODELS Definition: A structural causal model is a 4-tuple V,U, F, (u, where V {V 1,...,V n } are observable variables U {U 1,...,U m } are background variables F {f 1,..., f n } are functions determining V, v i f i (v, u (u is a distribution over U (u and F induce a distribution (v over observable variables
15 STRUCTURAL MODELS AND CAUSAL DIAGRAMS The arguments of the functions v i f i (v,u define a graph v i f i (pa i,u i A i V \ V i U i U Example: rice Quantity equations in economics U 1 I W U 2 q p b 1 b 2 p q + + d d 1 i 2 + u w + 1 u Q A Q 2
16 STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action do(x sets X to constants x regardless of the factors which previously determined X. do(x replaces all functions f i determining X with the constant functions Xx, to create a mutilated model M x q p b 1 b 2 p q + + d d 1 i 2 + w u + 1 u 2 U 1 I W U 2 Q
17 q p p STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action do(x sets X to constants x regardless of the factors which previously determined X. do(x replaces all functions f i determining X with the constant functions Xx, to create a mutilated model M x M p b 1 p b 2 q p + + d d 1 i 2 + w u + 1 u 2 U 1 I W U 2 Q p
18 CAUSAL MODELS AND COUNTERFACTUALS Definition: The sentence: Y would be y (in situation u, had X been x, denoted Y x (u y, means: The solution for Y in a mutilated model M x, (i.e., the equations for X replaced by X x with input Uu, is equal to y. Joint probabilities of counterfactuals: ( Y x y, Z w z u : Y x ( u y, Z w ( u z ( The super-distribution * is derived from M. arsimonous, consistent, and transparent u
19 AXIOMS OF CAUSAL COUNTERFACTUALS Y x ( u y : Y would be y, had X been x (in state U u 1. Definiteness x X s. t. X y ( u 2. Uniqueness 3. Effectiveness X xw ( u 4. Composition ( X y ( u x & ( X y ( u x ' x x x W x ( u w Y xw ( u Y x ( u 5. Reversibility ( Y xw ( u y & ( W xy ( u w Y x ( u x ' y
20 GRAHICAL COUNTERFACTUALS SYMBIOSIS Every causal graph expresses counterfactuals assumptions, e.g., X Y Z 1. Missing arrows Y Z Y x x, z ( u Y ( u 2. Missing arcs Y Z Y x Z y consistent, and readable from the graph. Every theorem in SEM is a theorem in N-R, and conversely.
21 STRUCTURAL ANALYSIS: SOME USEFUL RESULTS 1. Complete formal semantics of counterfactuals 2. Transparent language for expressing assumptions 3. Complete solution to causal-effect identification 4. Legal responsibility (bounds 5. Non-compliance (universal bounds 6. Integration of data from diverse sources 7. Direct and Indirect effects, 8. Complete criterion for counterfactual testability
22 REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY Regression (claimless: Y ax + a 1 z 1 + a 2 z a k z k + ε y a E[Y x,z] / x R yx z Y ε y X,Z Structural (empirical, falsifiable: Y bx + b 1 z 1 + b 2 z b k z k + u y b E[Y do(x,z] / x E[Y x,z ] / x Assumptions: Y x,z Y x,z,w cov(u i, u j for some i,j
23 REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY Regression (claimless: Y ax + a 1 z 1 + a 2 z a k z k + ε y a E[Y x,z] / x R yx z Y ε y X,Z Structural (empirical, falsifiable: Y bx + b 1 z 1 + b 2 z b k z k + u y b E[Y do(x,z] / x E[Y x,z ] / x Assumptions: Y x,z Y x,z,w cov(u i, u j for some i,j The mother of all questions: When would b equal a?, or What kind of regressors should Z include? Answer: When Z satisfies the backdoor criterion Question: When is b estimable by regression methods? Answer: graphical criteria available
24 THE BACK-DOOR CRITERION Graphical test of identification (y do(x is identifiable in G if there is a set Z of variables such that Z d-separates X from Y in G x. G Z 1 Z Z 2 1 G x Z Z 2 Z 3 Z 4 X Y Z 6 Z 5 Moreover, (y do(x (y x,z (z z ( adjusting for Z Z 3 Z 4 X Y Z 6 Z 5
25 RECENT RESULTS ON IDENTIFICATION do-calculus is complete Complete graphical criterion for identifying causal effects (Shpitser and earl, 26. Complete graphical criterion for empirical testability of counterfactuals (Shpitser and earl, 27.
26 DETERMINING THE CAUSES OF EFFECTS (The Attribution roblem Your Honor! My client (Mr. A died BECAUSE he used that drug.
27 DETERMINING THE CAUSES OF EFFECTS (The Attribution roblem Your Honor! My client (Mr. A died BECAUSE he used that drug. Court to decide if it is MORE ROBABLE THAN NOT that A would be alive BUT FOR the drug! N (? A is dead, took the drug >.5
28 THE ROBLEM Semantical roblem: 1. What is the meaning of N(x,y: robability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.
29 THE ROBLEM Semantical roblem: 1. What is the meaning of N(x,y: robability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. Answer: N ( x, y ( Y x ' y ' x, y Computable from M
30 THE ROBLEM Semantical roblem: 1. What is the meaning of N(x,y: robability that event y would not have occurred if it were not for event x, given that x and y did in fact occur. Analytical roblem: 2. Under what condition can N(x,y be learned from statistical data, i.e., observational, experimental and combined.
31 TYICAL THEOREMS (Tian and earl, 2 Bounds given combined nonexperimental and experimental data max 1 ( y ( y x' ( y' N min x' ( x,y ( x,y Identifiability under monotonicity (Combined data ( y x ( y x ( y x' ( y x' ( x,y N + x' ( y corrected Excess-Risk-Ratio
32 CAN FREQUENCY DATA DECIDE CAN FREQUENCY DATA DECIDE LEGAL RESONSIBILITY? Experimental Nonexperimental do(x do(x x x Deaths (y Survivals (y , 1, 1, 1, Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival laintiff: 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? N ( Y x ' y ' x, y >.5
33 SOLUTION TO THE ATTRIBUTION ROBLEM WITH ROBABILITY ONE 1 (y x x,y 1 Combined data tell more that each study alone
34 CONCLUSIONS Structural-model semantics, enriched with logic and graphs, provides: Complete formal basis for the N-R model Unifies the graphical, potential-outcome and structural equation approaches owerful and friendly causal calculus (best features of each approach
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