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THE MATHEMATICS OF CAUSAL INFERENCE IN STATISTICS Judea Pearl Department of Computer Science UCLA

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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 OUTLINE

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TRADITIONAL STATISTICAL INFERENCE PARADIGM Data Inference Q(P) (Aspects of P ) P Joint Distribution e.g., Infer whether customers who bought product A would also buy product B. Q = P(B | A)

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What happens when P changes? e.g., Infer whether customers who bought product A would still buy A if we were to double the price. FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Probability and statistics deal with static relations Data Inference Q( P ) (Aspects of P ) P Joint Distribution P Joint Distribution change

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FROM STATISTICAL TO CAUSAL ANALYSIS: 1. THE DIFFERENCES Note: P (v) P (v | price = 2) P does not tell us how it ought to change e.g. Curing symptoms vs. curing diseases e.g. Analogy: mechanical deformation What remains invariant when P changes say, to satisfy P (price=2)=1 Data Inference Q( P ) (Aspects of P ) P Joint Distribution P Joint Distribution change

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

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CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility 1.Causal and statistical concepts do not mix 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 + data causal assumptions causal conclusions }

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4.Non-standard mathematics: a)Structural equation models (Wright, 1920; Simon, 1960) b)Counterfactuals (Neyman-Rubin (Y x ), Lewis (x Y)) CAUSAL Spurious correlation Randomization Confounding / Effect Instrument Holding constant Explanatory variables STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility 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 + data causal assumptions causal conclusions }

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TWO PARADIGMS FOR CAUSAL INFERENCE Observed: P(X, Y, Z,...) Conclusions needed: P(Y x =y), P(X y =x | Z=z)... 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 P*(X, Y,…, Y x, X z,…) X, Y, Z constrain Y x, Z y,… Structural model Counterfactuals are derived quantities Subscripts modify the distribution P(Y x =y) = P x (Y=y)

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SUPER DISTRIBUTION IN N-R MODEL X X Y Y Y x= Z Z Y x= X z= X z= X y= Uu1 u2 u3 u4 Uu1 u2 u3 u4 inconsistency: x = 0 Y x=0 = Y Y = xY 1 + (1-x) Y 0

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Is consistency the only connection between Y x X | Z judgmental & opaque Try it: X Y Z ? opaque Y x X | Z judgmental & TYPICAL INFERENCE IN N-R MODEL Find P*(Y x =y) given covariate Z, Problems: 1) X, Y and Y x ? 2) Assume consistency: X=x Y x =Y Assume ignorability: Y x X | Z Is consistency the only connection between

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Data Inference Q(M) (Aspects of M ) Data Generating Model 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. Joint Distribution THE STRUCTURAL MODEL PARADIGM

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Z Y X INPUTOUTPUT FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION

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STRUCTURAL CAUSAL MODELS Definition: A structural causal model is a 4-tuple V,U, F, P(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) P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variables

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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 ) PA i V \ V i U i U Example: Price – Quantity equations in economics U1U1 U2U2 IW Q P PA Q

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U1U1 U2U2 IW Q P 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 X=x, to create a mutilated model M x STRUCTURAL MODELS AND INTERVENTION

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U1U1 U2U2 IW Q P P = p 0 MpMp 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 X=x, to create a mutilated model M x STRUCTURAL MODELS AND INTERVENTION

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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 U=u, is equal to y. Joint probabilities of counterfactuals: The super-distribution P * is derived from M. Parsimonous, consistent, and transparent

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AXIOMS OF CAUSAL COUNTERFACTUALS 1.Definiteness 2.Uniqueness 3.Effectiveness 4.Composition 5.Reversibility Y would be y, had X been x (in state U = u )

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GRAPHICAL – COUNTERFACTUALS SYMBIOSIS Every causal graph expresses counterfactuals assumptions, e.g., X Y Z consistent, and readable from the graph. Every theorem in SEM is a theorem in N-R, and conversely. 1.Missing arrows Y Z 2. Missing arcs YZ

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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

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REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY) Regression (claimless): Y = ax + a 1 z 1 + a 2 z a k z k + y a E[Y | x,z] / x = R yx z Y 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 ) = 0 for some i,j = =

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REGRESSION VS. STRUCTURAL EQUATIONS (THE CONFUSION OF THE CENTURY) Regression (claimless): Y = ax + a 1 z 1 + a 2 z a k z k + y a E[Y | x,z] / x = R yx z Y 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 ) = 0 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

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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 G x. Z6Z6 Z3Z3 Z2Z2 Z5Z5 Z1Z1 X Y Z4Z4 Z6Z6 Z3Z3 Z2Z2 Z5Z5 Z1Z1 X Y Z4Z4 Z Moreover, P(y | do(x)) = P(y | x,z) P(z) (adjusting for Z ) z GxGx G

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do -calculus is complete Complete graphical criterion for identifying causal effects (Shpitser and Pearl, 2006). Complete graphical criterion for empirical testability of counterfactuals (Shpitser and Pearl, 2007). RECENT RESULTS ON IDENTIFICATION

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DETERMINING THE CAUSES OF EFFECTS (The Attribution Problem) Your Honor! My client (Mr. A) died BECAUSE he used that drug.

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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 PN =

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THE PROBLEM Semantical Problem: 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.

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THE PROBLEM Semantical Problem: 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: Computable from M

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THE PROBLEM Semantical Problem: 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. Analytical Problem:

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TYPICAL THEOREMS (Tian and Pearl, 2000) Bounds given combined nonexperimental and experimental data Identifiability under monotonicity (Combined data) corrected Excess-Risk-Ratio

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CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY? Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival ExperimentalNonexperimental do(x) do(x ) x x Deaths (y) Survivals (y ) ,0001,0001,0001,000 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? Plaintiff: Mr. A is special.

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SOLUTION TO THE ATTRIBUTION PROBLEM WITH PROBABILITY ONE 1 P(y x | x,y) 1 Combined data tell more that each study alone

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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 Powerful and friendly causal calculus (best features of each approach)

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