CAUSAL INFERENCE AS COMPUTATIONAL LEARNING. Judea Pearl University of California Los Angeles (

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1 CAUSAL INFERENCE AS COMUTATIONAL LEARNING Judea earl University of California Los Angeles

2 OUTLINE Inference: Statistical vs. Causal distinctions and mental barriers Formal semantics for counterfactuals: definition, axioms, graphical representations Inference to three types of claims: 1. Effect of potential interventions 2. Attribution Causes of Effects 3. Direct and indirect effects

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 Inference Joint Distribution Q Aspects of 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 price=2=1 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 ropensity score 3. 4.

7 FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS 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 assumptions + data causal assumptions } STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility ropensity score causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics.

8 FROM STATISTICAL TO CAUSAL ANALYSIS: 2. MENTAL BARRIERS 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 assumptions + data causal assumptions } STATISTICAL Regression Association / Independence Controlling for / Conditioning Odd and risk ratios Collapsibility ropensity score causal conclusions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. Non-standard mathematics: a Structural equation models Wright, 1920; Simon, 1960 b Counterfactuals Neyman-Rubin Y x, Lewis x Y

9 WHY CAUSALITY NEEDS SECIAL MATHEMATICS Scientific Equations e.g., Hooke s Law are non-algebraic e.g., ricing olicy: Double the competitor s price Correct notation: Y := = 2X X = 1 rocess information X = 1 Y = 2 The solution Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must wipe out X = 1.

10 WHY CAUSALITY NEEDS SECIAL MATHEMATICS Scientific Equations e.g., Hooke s Law are non-algebraic e.g., ricing olicy: Double the competitor s price Correct notation: or Y 2X X = 1 rocess information Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must wipe out X = 1. X = 1 Y = 2 The solution

11 THE STRUCTURAL MODEL ARADIGM Data Joint Distribution Data Generating Model QM Aspects of M M Inference M Invariant strategy mechanism, recipe, law, protocol by which Nature assigns values to variables in the analysis.

12 FAMILIAR CAUSAL MODEL ORACLE FOR MANIILATION X Y Z INUT OUTUT

13 STRUCTURAL CAUSAL MODELS Definition: A structural causal model is a 4-tuple ÆV,U, F, uæ, 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

14 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 U 2 I W Q A Q u w d q b p u i d p b q + + = + + =

15 STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action dox sets X to constants x regardless of the factors which previously determined X. dox replaces all functions f i determining X with the constant functions X=x, to create a mutilated model M x q p = = b1 p b2q + + d1i + u1 d2w + u2 U 1 I W U 2 Q

16 STRUCTURAL MODELS AND INTERVENTION Let X be a set of variables in V. The action dox sets X to constants x regardless of the factors which previously determined X. dox replaces all functions f i determining X with the constant functions X=x, to create a mutilated model M x M p q p p = = = b1 p b2q p0 + + d1i + u1 d2w + u2 U 1 I W U 2 Q = p 0

17 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. The Fundamental Equation of Counterfactuals: Yx u = Y u M x

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 U=u, is equal to y. u Y u Y M x x = The Fundamental Equation of Counterfactuals:,, : u z Z y Y z u Z y u u Y w x w x = = = = = Joint probabilities of counterfactuals:,, ' ' : ' : ' y x u y x y Y N u y Y y y u Y u x y u Y u x x x = = = = = = = In particular: dox

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 = x 2. Uniqueness X y u = x & X y u = x' x = x' 3. Effectiveness X xw u = x 4. Composition W x u = w Yxw u = Yx u 5. Reversibility Yxw u = y & Wxy u = w Yx u = y

20 INFERRING THE EFFECT OF INTERVENTIONS The problem: To predict the impact of a proposed intervention using data obtained prior to the intervention. The solution conditional: Causal Assumptions + Data olicy Claims 1. Mathematical tools for communicating causal assumptions formally and transparently. 2. Deciding mathematically whether the assumptions communicated are sufficient for obtaining consistent estimates of the prediction required Deriving Suggesting if 2 if 2 is affirmative is negative a closed-form set of measurements expression and for experiments the predicted that, impact if performed, would render a consistent estimate feasible.

21 NON-ARAMETRIC STRUCTURAL MODELS Given x,y,z, should we ban smoking? Smoking a U1 U 2 X Z Y Tar in Lungs Linear Analysis x = u 1, z = αx + u 2, y = βz + γ u 1 + u 3. b Cancer U 3 f 1 f2 Smoking U1 U 2 X Z Y Tar in Lungs f 3 Cancer U 3 Nonparametric Analysis x = f 1 u 1, z = f 2 x, u 2, y = f 3 z, u 1, u 3. Find: α β Find: y dox

22 Given x,y,z, should we ban smoking? Smoking Linear Analysis x = u 1, z = αx + u 2, y = βz + γ u 1 + u 3. Find: α β EFFECT OF INTERVENTION a U1 U 2 X Z Y Tar in Lungs b AN EXAMLE Cancer U 3 U2 f 3 Smoking Find: y dox = Y=y in new model f 2 U1 X = x Z Y Tar in Lungs Cancer U 3 Nonparametric Analysis x = const. z = f 2 x, u 2, y = f 3 z, u 1, u 3.

23 EFFECT OF INTERVENTION AN EXAMLE cont Given x,y,z, should we ban smoking? U unobserved U unobserved Smoking X Z Y Tar in Lungs Cancer Smoking X = x Z Y Tar in Lungs Cancer

24 EFFECT OF INTERVENTION AN EXAMLE cont Given x,y,z, should we ban smoking? U unobserved U unobserved Smoking X Z Y Tar in Lungs Cancer Smoking X = x Z Y Tar in Lungs Cancer re-intervention ost-intervention x, y, z = u x u z x y z, u y, z do x = u z x y z, u u u

25 EFFECT OF INTERVENTION AN EXAMLE cont Given x,y,z, should we ban smoking? U unobserved U unobserved Smoking X Z Y Tar in Lungs Cancer Smoking X = x Z Y Tar in Lungs Cancer re-intervention ost-intervention x, y, z = u x u z x y z, u y, z do x = u z x y z, u u To compute y,z dox, we must eliminate u. graphical problem. u

26 ELIMINATING CONFOUNDING BIAS A GRAHICAL CRITERION y dox is estimable if there is a set Z of variables such that Z d-separates X from Y in G x. G G x Z 1 Z 2 Z 1 Z Z 2 Z 3 Z 3 Z 4 Z 5 Z 4 Z 5 X Z 6 Y X Z 6 Y Moreover, y dox = y x,z z z adjusting for Z

27 RULES OF CAUSAL CALCULUS Rule 1: Ignoring observations y do{x}, z, w = y do{x}, w Rule 2: Action/observation exchange y do{x}, do{z}, w = y do{x},z,w Rule 3: Ignoring actions Y Z X,W y do{x}, do{z}, w = y do{x}, w if if if Y Z X,W Y Z X,W G X G X Z G X ZW

28 DERIVATION IN CAUSAL CALCULUS Genotype Unobserved Smoking Tar Cancer c do{s} = Σ t c do{s}, t t do{s} robability Axioms = S t c do{s}, do{t} t do{s} = S t c do{s}, do{t} t s = S t c do{t} t s = S s S t c do{t}, s s do{t} t s = S s S t c t, s s do{t} t s = S s S t c t, s s t s Rule 2 Rule 2 Rule 3 robability Axioms Rule 2 Rule 3

29 INFERENCE ACROSS DESIGNS roblem: redict y dox from a study in which only Z can be controlled. Solution: Determine if y dox can be reduced to a mathematical expression involving only doz.

30 COMLETENESS RESULTS ON IDENTIFICATION do-calculus is complete Complete graphical criterion for identifying causal effects Shpitser and earl, Complete graphical criterion for empirical testability of counterfactuals Shpitser and earl, 2007.

31 THE CAUSAL RENAISSANCE: From Hoover 2004 Lost Causes VOCABULARY IN ECONOMICS From Hoover 2004 Lost Causes

32 THE CAUSAL RENAISSANCE: 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. Imperfect experiments universal bounds for IV 6. Integration of data from diverse sources 7. Direct and Indirect effects, 8. Complete criterion for counterfactual testability

33 EFFECT DECOMOSITION direct vs. indirect effects 1. Why decompose effects? 2. What is the semantics of direct and indirect effects? 3. What are the policy implications of direct and indirect effects? 4. When can direct and indirect effect be estimated consistently from experimental and nonexperimental data?

34 WHY DECOMOSE EFFECTS? 1. To understand how Nature works 2. To comply with legal requirements 3. To predict the effects of new type of interventions: Signal routing, rather than variable fixing

35 LEGAL IMLICATIONS OF DIRECT EFFECT Can data prove an employer guilty of hiring discrimination? Gender X Z Qualifications What is the direct effect of X on Y? averaged over z Y Hiring E Y do x1, do z E Y do x0, do z Adjust for Z? No! No!

36 NATURAL SEMANTICS OF AVERAGE DIRECT EFFECTS Robins and Greenland 1992 ure X Z z = f x, u y = g x, z, u Y Average Direct Effect of X on Y: DE x0, x1; Y The expected change in Y, when we change X from x 0 to x 1 and, for each u, we keep Z constant at whatever value it attained before the change. E[ Y x Y 1Zx x 0 0 In linear models, DE = Controlled Direct Effect ]

37 SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS Consider the quantity Given M, u, Q is well defined u] Given u, Z x *u is the solution for Z in M x *, call it z Yx Zx is the solution for Y in M * u u xz experiment al Can Q be estimated from data? nonexperim ental Experimental: nest-free expression Nonexperimental: subscript-free expression Q = E u [ Y x Zx * u

38 NATURAL SEMANTICS OF INDIRECT EFFECTS X Z z = f x, u y = g x, z, u Y IE 1 Indirect Effect of X on Y: The expected change in Y when we keep X constant, say at x 0, and let Z change to whatever value it would have attained had X changed to x 1. E[ Y x Y x0, x ; Y 0Zx x 1 0 In linear models, IE = TE - DE ]

39 OLICY IMLICATIONS OF INDIRECT EFFECTS What is the indirect effect of X on Y? The effect of Gender on Hiring if sex discrimination is eliminated. GENDER X Z QUALIFICATION IGNORE f Y HIRING Blocking a link a new type of intervention

40 RELATIONS BETWEEN TOTAL, DIRECT, AND INDIRECT EFFECTS Theorem 5: The total, direct and indirect effects obey The following equality TE x, x*; Y = DE x, x*; Y IE 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 *.

41 EXERIMENTAL IDENTIFICATION OF AVERAGE DIRECT EFFECTS Theorem: If there exists a set W such that Yxz Zx* W for all z and x Then the average direct effect DE x x*; Y = E Y,, x Z E Y * x* x Is identifiable from experimental data and is given by [ E Y w E Y w ] DE x, x*; Y = xz x* z Zx* = z w w w, z

42 Theorem: If there exists a set W such that then, DE x, x*; Y Example: GRAHICAL CONDITION FOR EXERIMENTAL IDENTIFICATION OF DIRECT EFFECTS Y Z W G and W ND X Z = XZ [ E Y w E Y w ] xz x* z Zx* = z w w w, z

43 GENERAL ATH-SECIFIC EFFECTS Def. X x * X W Z W Z z * = Z x* u Y Y M g * i i = Form a new model,, specific to active subgraph g f * pa, u; g f pa g, pa* g, u Definition: g-specific effect Nonidentifiable even in Markovian models i E x, x* ; Y = TE x, x* ; Y g M i i M * g

44 SUMMARY OF RESULTS 1. Formal semantics of path-specific effects, based on signal blocking, instead of value fixing. 2. ath-analytic techniques extended to nonlinear and nonparametric models. 3. Meaningful graphical conditions for estimating direct and indirect effects from experimental and nonexperimental data.

45 CONCLUSIONS Structural-model semantics, enriched with logic and graphs, provides: Complete formal basis for causal and counterfactual reasoning Unifies the graphical, potential-outcome and structural equation approaches rovides friendly and formal solutions to century-old problems and confusions.

46

47 WHY CAUSALITY NEEDS SECIAL MATHEMATICS SEM Equations are Non-algebraic e.g., ricing olicy: Double the competitor s price Correct notation: Y := = 2X X = 1 rocess information X = 1 Y = 2 Static information Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must wipe out X = 1.

48 ELIMINATING CONFOUNDING BIAS A GRAHICAL CRITERION y dox is estimable if there is a set Z of variables such that Z d-separates X from Y in G x. G G x Z 1 Z 2 Z 1 Z Z 2 Z 3 Z 3 Z 4 Z 5 Z 4 Z 5 X Z 6 Y X Z 6 Y Moreover, y dox = y x,z z z adjusting for Z

49 RULES OF CAUSAL CALCULUS Rule 1: Ignoring observations y do{x}, z, w = y do{x}, w Rule 2: Action/observation exchange y do{x}, do{z}, w = y do{x},z,w Rule 3: Ignoring actions Y Z X,W y do{x}, do{z}, w = y do{x}, w if if if Y Z X,W Y Z X,W G X G X Z G X ZW

50 DERIVATION IN CAUSAL CALCULUS Genotype Unobserved Smoking Tar Cancer c do{s} = Σ t c do{s}, t t do{s} robability Axioms = S t c do{s}, do{t} t do{s} = S t c do{s}, do{t} t s = S t c do{t} t s = S s S t c do{t}, s s do{t} t s = S s S t c t, s s do{t} t s = S s S t c t, s s t s Rule 2 Rule 2 Rule 3 robability Axioms Rule 2 Rule 3

51 INFERENCE ACROSS DESIGNS roblem: redict y dox from a study in which only Z can be controlled. Solution: Determine if y dox can be reduced to a mathematical expression involving only doz.

52 COMLETENESS RESULTS ON IDENTIFICATION do-calculus is complete Complete graphical criterion for identifying causal effects Shpitser and earl, Complete graphical criterion for empirical testability of counterfactuals Shpitser and earl, 2007.

53 THE CAUSAL RENAISSANCE: From Hoover 2004 Lost Causes VOCABULARY IN ECONOMICS From Hoover 2004 Lost Causes

54 THE CAUSAL RENAISSANCE: 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. Imperfect experiments universal bounds for IV 6. Integration of data from diverse sources 7. Direct and Indirect effects, 8. Complete criterion for counterfactual testability

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

56 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 > 0.50

57 THE ROBLEM Semantical roblem: 1. What is the meaning of Nx,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.

58 THE ROBLEM Semantical roblem: 1. What is the meaning of Nx,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 = Yx ' = y' x, y Computable from M

59 THE ROBLEM Semantical roblem: 1. What is the meaning of Nx,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 Nx,y be learned from statistical data, i.e., observational, experimental and combined.

60 TYICAL THEOREMS Tian and earl, 2000 Bounds given combined nonexperimental and experimental data 1 min 0 max x,y y' N x,y y y x' x' x,y y x' y x y x' y x y N x' + = Identifiability under monotonicity Combined data corrected Excess-Risk-Ratio

61 CAN FREQUENCY DATA DECIDE LEGAL RESONSIBILITY? Experimental Nonexperimental dox dox x x Deaths y Survivals y ,000 1,000 1,000 1,000 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 = Yx ' = y' x, y > 0.50

62 TYICAL THEOREMS Tian and earl, 2000 Bounds given combined nonexperimental and experimental data 1 min 0 max x,y y' N x,y y y x' x' x,y y x' y x y x' y x y N x' + = Identifiability under monotonicity Combined data corrected Excess-Risk-Ratio

63 SOLUTION TO THE ATTRIBUTION ROBLEM WITH ROBABILITY ONE 1 y x x,y 1 Combined data tell more that each study alone

64 LEGAL DEFINITION OF DIRECT EFFECT 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 FE Cases 921, 7 th Cir. 1996] x = male, x = female y = hire, y = not hire z = applicant s qualifications NO DIRECT EFFECT Y x ' = Y Z x x