THE SCIENCE OF CAUSE AND EFFECT

Transcription

1 HE CIENCE OF CAUE AND EFFEC Judea Pearl University of California Los Angeles ( OULINE 1. he causal revolution from associations to intervention to counterfactuals 2. he two fundamental laws of causal inference 3. From counterfactuals to problem solving a) policy evaluation (AE, E, ) b) ediation c) transportability external validity d) missing data e) [attribution, selection bias, heterogeneity] RADIIONAL AIICAL INFERENCE PARADIG 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) 1

2 FRO AOCIAION O INERVENION Data P Joint Distribution change Inference P Joint Distribution Q(P ) (Aspects of P ) e.g., Estimate P (sales) if we double the price. How does P change to P? New oracle e.g., Estimate P (cancer) if we ban smoking. FRO AOCIAION O COUNERFACUAL: Probability and statistics deal with static relations Data P Joint Distribution change Inference P Joint Distribution Q(P ) (Aspects of P ) hat happens when P changes? e.g., Estimate the probability that a customer who bought A would buy A if we were to double the price. HE RUCURAL ODEL PARADIG Data Joint Distribution Data Generating odel Q() (Aspects of ) Inference Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis. A painful de-crowning of a beloved oracle! 2

3 FRO AIICAL O CAUAL ANALI: HE HARP BOUNDAR 1. Causal and associational concepts do not mix. CAUAL AOCIAIONAL purious correlation Regression Randomization / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odds and risk ratios Ignorability / Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score FRO AIICAL O CAUAL ANALI: 3. HE ENAL BARRIER 1. Causal and associational concepts do not mix. CAUAL AOCIAIONAL purious correlation Regression Randomization / Intervention Association / Independence Confounding / Effect Controlling for / Conditioning Instrumental variable Odds and risk ratios Ignorability / Exogeneity Collapsibility / Granger causality Explanatory variables Propensity score 2. No causes in no causes out (Cartwright, 1989) data causal assumptions 3. Causal assumptions cannot be expressed in the mathematical language of standard statistics. 4. Non-standard mathematics: a) tructural equation models (right, 1920; imon, 1960) b) Counterfactuals (Neyman-Rubin ( x ), Lewis (x )) } causal conclusions 3

4 HE NE ORACLE: RUCURAL CAUAL ODEL: HE ORLD A A COLLECION OF PRING Definition: A structural causal model is a 4-tuple <V,U, F, P(u)>, where V = {V 1,...,V n } are endogenous variables U = {U 1,...,U m } are background variables F = {f 1,..., f n } are functions determining V, v i = f i (v, u) e.g., y = α + βx + u Not regression!!!! P(u) is a distribution over U P(u) and F induce a distribution P(v) over observable variables COUNERFACUAL ARE EBARRAINGL IPLE Definition: x (u): hat would be had been x. x (u) = the solution for in a mutilated model x, in which the equation for is replaced by = x. U x U (u) (u) = x x (u) he Fundamental Equation of Counterfactuals: x (u) Δ = x (u) EFFEC OF INERVENION ARE EBARRAINGL IPLE Definition: he effect of setting to x, P( = y do (=x)), is equal to the probability of = y in a mutilated model x, in which the equation for is replaced by = x. U x U (u) (u) = x x (u) he Fundamental Equation of Interventions: P( = y do( = x)) Δ = P x ( = y) = P( x = y) 4

5 EIAING HE EFFEC OF INERVENION IHOU EQUAION U x U (u) (u) = x x (u) he Fundamental Equation of Interventions: P( = y do( = x)) Δ = P ( = y) x P(x, y,u) = P(u)P(x u)p(y x,u) P(y,u do(x)) = P(u)P(y x,u) runcated product P(y do(x)) = P(y x,u)p(u) Adjustment formula u READING COUNERFACUAL FRO E ε 1 ε 2 β γ ε3 = 2.0 γ = 0.4 ε3 α α = 0.7 = 2.3 = reatment = tudy ime x = ε 1 z = βx + ε 2 z = 2.0 = core y = αx + γz + ε 3 Data shows: α = 0.7, β = 0.5, γ = 0.4 A student named Joe, measured = 0.5, = 1.0, = 1.9 Q 1 : hat would Joe s score be had he doubled his study time? Answer: =2 = ε 3 = 2.30 ε 2 β = 0.5 ε 1 HE O FUNDAENAL LA OF CAUAL INFERENCE 1. he Law of Counterfactuals (and Interventions) x (u) = x (u) ( generates and evaluates all counterfactuals.) 2. he Law of Conditional Independence (d-separation) ( sep ) G( ) ( ) P(v) (eparation in the model independence in the distribution.) 5

6 HE LA OF CONDIIONAL INDEPENDENCE Graph (G) (prinkler) C (Climate) (etness) R (Rain) odel () C = f C (U C ) = f (C,U ) R = f R (C,U R ) = f (, R,U ) Gift of the Gods If the U 's are independent, the observed distribution P(C,R,,) satisfies constraints that are: (1) independent of the f 's and of P(U), (2) readable from the graph. D-EPARAION: NAURE LANGUAGE FOR COUNICAING I RUCURE Graph (G) (prinkler) C (Climate) (etness) R (Rain) odel () C = f C (U C ) = f (C,U ) R = f R (C,U R ) = f (, R,U ) Every missing arrow advertises an independency, conditional on a separating set. e.g., C (, R) R C Applications: 1. odel testing 2. tructure learning 3. Reducing "what if I do" questions to symbolic calculus 4. Answering scientific questions from the graph HA IF VARIABLE ARE UNOBERVED? EFFEC OF AR-UP ON INJUR (hrier & Platt, 2008) AE = E = PNC = No, no! 6

7 AHEAICAL REUL #1: (Intervention is a solved problem) he estimability of any expression of the form Q = P(y 1, y 2, y 3,..., y m do(x 1, x 2,..., x n ), 1, 2,..., k ) Can be determined in polynomial time, given any causal graph G with both measured and unmeasured variables. If Q is estimable, then its estimand can be derived in polynomial time he algorithm is complete OULINE 1. he causal revolution from associations to intervention to counterfactuals 2. he two fundamental laws of causal inference 3. From counterfactuals to problem solving a) policy evaluation (AE, E, ) b) ediation c) transportability external validity d) missing data e) [attribution, selection bias, heterogeneity] EDIAION: A COUNERFACUAL RIUPH 1. hy decompose effects? 2. hat is the definition of direct and indirect effects? 3. hat are the policy implications of direct and indirect effects? 4. hen can direct and indirect effect be estimated consistently from experimental and nonexperimental data? 21 7

8 H DECOPOE EFFEC? 1. o understand how Nature works 2. o comply with legal requirements 3. o predict the effects of new type of interventions: ignal re-routing and mechanism deactivating, rather than variable fixing 22 LEGAL IPLICAION OF DIREC EFFEC Can data prove an employer guilty of hiring discrimination? (Gender) (Qualifications) (Hiring) hat is the direct effect of on? (CDE) E( do(x 1 ),do(z)) E( do(x 0 ),do(z)) (z-dependent) Adjust for? No! No! Identification is completely solved (ian & hpiser, 2006) 23 RADIIONAL EDIAION ANALI A CURABLE BAD HABI 1. o prevent from varying, control for, the resulting partial regression would be the direct effect. 2. rong! Controlling does not prevent from varying. 3. Example: fix ( = 0) 1 L L a b c = + L, = L controlling for = 0 yields = - disabled disabled Fixing = 0 yields = L independent of "he best way to discuss moderation or mediation is to set aside the entire literature on these topics and start from scratch." (Rod cdonald, 2001) 8

9 NAURAL INERPREAION OF AVERAGE DIREC EFFEC Robins and Greenland (1992), Pearl (2001) z = f (x, u) y = g (x, z, u) Natural Direct Effect of on : DE(x 0, x 1 ; ) he expected change in, when we change from x 0 to x 1 and, for each u, we keep constant at whatever value it attained before the change. E[ x 1x0 x0 ] In linear models, DE = Controlled Direct Effect = β(x 1 x 0 ) DEFINIION OF INDIREC EFFEC z = f (x, u) y = g (x, z, u) Indirect Effect of on : IE(x 0, x 1 ; ) he expected change in when we keep constant, say at x 0, and let change to whatever value it would have attained had changed to x 1. E[ x 0 x1 x0 ] In linear models, IE = E - DE No controlled indirect effect 26 POLIC IPLICAION OF INDIREC EFFEC hat is the indirect effect of on? he effect of Gender on Hiring if sex discrimination is eliminated. GENDER QUALIFICAION IGNORE f HIRING Deactivating a link a new type of intervention 27 9

10 HE EDIAION FORULA IN UNCONFOUNDED ODEL z = f (x, u 1 ) y = g (x, z, u 2 ) u 1 independent of u 2 DE = [E( x 1,z) E( x 0,z)]P(z x 0 ) z IE = [E( x 0,z)[P(z x 1 ) P(z x 0 )] z E = E( x 1 ) E( x 0 ) E DE + IE IE = Fraction of responses explained by mediation Complete (sufficient) identification conditions for confounded models with multiple mediators (Pearl 2001; hpitser 2013). E DE = Fraction of responses owed to mediation (necessary) HA CAN EDIAION FORULA DO FOR PARAERIC ANAL? ulti-mediators non-linear models γ α 2 β 4 γ β 1 1 β 2 β 3 y = β 1 m + β 2 x + β 3 xm + β 4 w + u 1 m = γ 1 x + γ 2 w + u 2 w = αx + u 3 hat combination of parameters gives the effect mediated by? IE( ) = β 1 (γ 1 + αγ 2 ) hat combination of parameters gives the effect owed to? E DE( ) = (β 1 + β 3 )(γ 1 + αγ 2 ) HEN CAN E IDENIF EDIAED EFFEC? (a) 2 3 (b) 2 3 (c) 2 3 (d) 2 3 (e) 2 3 (f)

11 HEN CAN E IDENIF EDIAED EFFEC? (a) 2 3 (b) 2 3 (c) 2 3 (d) 2 3 (e) 2 3 (f) AHEAICAL REUL #2: (Natural mediation is a solved problem) Ignorability is not required for identifying natural effects he nonparametric estimability of natural (and controlled) direct and indirect effects can be determined mechanically given any causal graph G with both measured and unmeasured variables. If NDE (or NIE) is estimable, then its estimand can be derived mechanically in polynomial time. he algorithm is complete and was extended to any path-specific effect by hpitser (2013). OULINE 1. he causal revolution from associations to intervention to counterfactuals 2. he two fundamental laws of causal inference 3. From counterfactuals to problem solving a) policy evaluation (AE, E, ) b) ediation c) transportability external validity d) missing data e) [attribution, selection bias, heterogeneity] 11

12 RANPORABILI OF KNOLEDGE ACRO DOAIN (with E. Bareinboim) A heory of Causal ransportability hen can causal relations learned from experiments be transferred to another environment, different from the first, in which no experiment can be conducted. External Validity Decades of Literature Cox (1958) Campbell and tanley (1963) anski (2007) OVING FRO HE LAB O HE REAL ORLD... Lab H 1 Real world Everything is assumed to be the same, trivially transportable! H 2 Everything is assumed to be the different, not transportable! REUL: ALGORIH O DEERINE IF AN EFFEC I RANPORABLE ' INPU: Annotated Causal Graph U Factors creating differences V OUPU: 1. ransportable or not? 2. easurements to be taken in the experimental study 3. easurements to be taken in the target population 4. A transport formula 5. Completeness (Bareinboim, 2012) P*(y do(x)) = P(y do(x),z) P *(z w) P(w do(w),t)p *(t) z w t 12

13 EA-NHEI A ORK arget population * (a) (b) R = P*(y do(x)) (c) (d) (e) (f) (g) (h) (i) AHEAICAL REUL #3: (ransportability and meta-transportability are solved) Nonparametric transportability of experimental results from multiple environments can be decided in polynomial time, provided commonalities and differences are encoded in selection diagrams. hen transportability is feasible, the transport formula can be derived in polynomial time, which specifies the information needed to be extracted from each environment to synthesize a consistent estimate for the target environment. he algorithm is complete. OULINE 1. he causal revolution from associations to intervention to counterfactuals 2. he two fundamental laws of causal inference 3. From counterfactuals to problem solving a) policy evaluation (AE, E, ) b) ediation c) transportability external validity d) missing data e) [attribution, selection bias, heterogeneity] 13

14 IING DAA: FRO A CAUAL INFERENCE PERPECIVE (ohan, Pearl & ian 2013) Pervasive in every experimental science. Huge literature, powerful software industry, deeply entrenched culture. Current practices are based on statistical characterization (Rubin, 1976) of a problem that is inherently causal. Needed: (1) theoretical guidance, (2) performance guarantees, and (3) tests of assumptions. Graphical GRAPHICAL odels for Inference ODEL ith FOR issing Data Karthika ohan, IING Judea Pearl DAA and Jin ian * R P(*,,,R ) m m m m Distribution with missing values Discomfort * reatment Observed proxy of Outcome R Cause of missingness in Graph depicting the missingness process HA CAN CAUAL HEOR DO FOR IING DAA? Q-1. hat should the world be like, for a given statistical procedure to produce the expected result? Q-2. Can we tell from the postulated world whether any method can produce a bias-free result? How? Q-3. Can we tell from data if the world does not work as postulated? o answer these questions, we need models of the world, i.e., process models. tatistical characterization of the problem is too crude, e.g., CAR, AR, NAR. testable untestable recoverable non-recoverable 14

15 RECOVERABILI AND EABILI Recoverability Given a missingness model G and data D, when is a quantity Q estimable from D without bias? Non-recoverability heoretical impediment to any estimation strategy estability Given a model G, when does it have testable implications (refutable by some partially-observed data D' )? hat is known about Recoverability and estability? CAR recoverable almost testable AR recoverable uncharted NAR uncharted uncharted I P(,) RECOVERABLE? P( ) R (a) R R R R (b) (c) R R P() (d) R R (e) R R P(,,) (f) R R R R (g) R R R (h) R R P(,,) (i) R HE RECOVERABILI PIE (and what s in it for the user) CAR AR equential-ar () () (N) arkovian* Others? { } = Non-recoverable (provable) Recoverability in AR and CAR models can be achieved by model-blind estimators (e.g., I or E). In areas () and (), recoverability requires model-smart estimators. estability charted over the entire terrain. 15

16 HE PECULIAR CHARACER OF EABILI IN IING DAA Not all that looks testable is testable: ome testable implications of fully recovered distributions are not testable from missing data. Example: (treatment) (discomfort) Now is testable R z (masking ) P(,, ) is recoverable, and advertises the conditional independence, which is falsifiable, hence testable. et is not falsifiable by any data in which is partially missing. Any such data, even when generated by a model in which is false, may be construed as if generated by the model above, in which is true. (outcome) AN IPOIBILI HEORE FOR IING DAA (a) Accident Injury (b) Injury reatment Education (latent) R x issing () wo statistically indistinguishable models, yet P(,) is recoverable in (a) and not in (b). No universal algorithm exists that decides recoverability (or guarantees unbiased results) without looking at the model. R x issing () A RONGER IPOIBILI HEORE (a) (b) R x R x wo statistically indistinguishable models, P() is recoverable in both, but through two different methods: In (a): P() = y P( )P(, R x = 0), while in (b): P() = P( R x = 0) No universal algorithm exists that produces an unbiased estimate whenever such exists. 16

17 HE PROBLE OF ELECION BIA ystematic exclusion of samples from the data is a major obstacle to valid causal and statistical inferences; In general, it cannot be removed by randomized experiments and can hardly be detected in either experimental or passive observations. Goal: Provide methods capable of mitigating and sometimes eliminating this bias. (Joint work with Bareinboim & ian) 49 APLE ELECION IN HE LANGUAGE OF GRAPH Augmented graph U c 0 β 1 β 2 U = 1: Included in the sample = 0: Excluded from the sample 50 O OURCE OF ELECION BIA: Collider U Virtual collider U c 0 β 1 β 2 U Cannot be eliminated by adjustment or by randomization 51 17

18 CONFOUNDING BIA vs ELECION BIA Unblockable flow of information between treatment and outcome spurious correlation. G1 G2 52 HE GENERAL ELECION BIA PROBLE Input: An augmented causal graph, describing a hypothesized selection process Under what conditions can we estimate the query (e.g., P(y x)) from P(v = 1)). 53 HE ELECION BIA PROBLE election bias, caused by preferential exclusion of samples from the data, is a major obstacle to valid causal and statistical inferences; Under what conditions can we estimate the query (e.g., P(y x)) from P(v = 1)

19 RECOVERING FRO ELECION BIA Question: Under what conditions can we estimate the distribution P(y x) from P(v =1). heorem: Q = P(y x) is recoverable from selection biased data if and only if ( ) G. P(y x) is not recoverable. P(y x) is recoverable. 55 ELECION IH EERNAL INFORAION Estimate Q = P(y x) from selection biased data Q is not recoverable by the previous theorem but what if P( 1, 2) is available? 56 RECOVERABILI IH EERNAL INFORAION heorem. P(y x) is recoverable if there is a set C such that ( C, ) holds in G and P(C, ) is estimable. oreover, P(y x) = P(y x, c, = 1) P(c x) c C = { 1, 2}? yes = { 2, 1, 2}? no = { 2, 3}? yes

20 UAR OF ELECION BIA REUL Nonparametric recoverability from selection bias can be decided provided that an augmented causal graph is available. hen recoverability is feasible, the estimand can be derived in polynomial time. he result is complete for pure recoverability and sufficient for recoverability with external information. he back-door criterion can be generalized to handle selection bias. tronger results can be obtained for the OR recoverability. CONCLUION 1. hink nature, not data, not even experiment. 2. hink hard, but only once the rest is mechanizable. 3. peak a language in which the veracity of each assumption can be judged by users, and which tells you whether any of those assumptions can be refuted by data. 4. Proceed in a language in which your research question can be answered from the assumptions plus the data. hank you 20

21 RANPORABILI OF KNOLEDGE ACRO DOAIN (with E. Bareinboim) 1. A heory of causal transportability hen can causal relations learned from experiments be transferred to a different environment in which no experiment can be conducted? 2. A heory of statistical transportability hen can statistical information learned in one domain be transferred to a different domain in which a. only a subset of variables can be observed? Or, b. only a few samples are available? OVING FRO HE LAB O HE REAL ORLD... Lab H 1 Real world Everything is assumed to be the same, trivially transportable! H 2 Everything is assumed to be the different, not transportable! OIVAION HA CAN EPERIEN IN LA ELL ABOU NC? (Age) (Age) * (Intervention) (Outcome) (Observation) (Outcome) Experimental study in LA easured: P(x, y,z) P(y do(x),z) P*(y do(x)) =? Observational study in NC easured: P*(x, y,z) P*(z) P(z) Needed: = P(y do(x),z)p*(z) z ransport Formula (calibration): F(P,P do,p*) 21

22 RANPOR FORULA DEPEND ON HE OR (a) a) represents age P*(y do(x)) = P(y do(x),z)p*(z) z b) represents language skill P*(y do(x)) =? P(y do(x)) (b) Factors producing differences RANPOR FORULA DEPEND ON HE OR (a) a) represents age P*(y do(x)) = P(y do(x),z)p*(z) z b) represents language skill P*(y do(x)) = P(y do(x)) c) represents a bio-marker P*(y do(x)) =? P(y do(x),z)p*(z x ) z (b) (c) GOAL: ALGORIH O DEERINE IF AN EFFEC I RANPORABLE ' U V INPU: Annotated Causal Graph Factors creating differences OUPU: 1. ransportable or not? 2. easurements to be taken in the experimental study 3. easurements to be taken in the target population 4. A transport formula P*(y do(x)) = f [P(y,v,z,w,t,u do(x)); P *(y,v,z,w,t,u)] 22

23 RANPORABILI REDUCED O CALCULU heorem A causal relation R is transportable from to * if and only if it is reducible, using the rules of do-calculus, to an expression in which is separated from do( ). R ( * )= P*(y do(x)) = P(y do(x),s) = P(y do(x),s,w)p(w do(x),s) w = P(y do(x),w)p(w s) w = P(y do(x),w)p*(w) w REUL: ALGORIH O DEERINE IF AN EFFEC I RANPORABLE ' INPU: Annotated Causal Graph U Factors creating differences V OUPU: 1. ransportable or not? 2. easurements to be taken in the experimental study 3. easurements to be taken in the target population 4. A transport formula 5. Completeness (Bareinboim, 2012) P*(y do(x)) = P(y do(x),z) P *(z w) P(w do(w),t)p *(t) z w t FRO EA-ANALI O EA-NHEI he problem How to combine results of several experimental and observational studies, each conducted on a different population and under a different set of conditions, so as to construct an aggregate measure of effect size that is "better" than any one study in isolation. 23

24 EA-NHEI A ORK arget population * (a) (b) R = P*(y do(x)) (c) (d) (e) (f) (g) (h) (i) EA-NHEI REDUCED O CALCULU heorem { 1, 2,, K } a set of studies. {D 1, D 2,, D K } selection diagrams (relative to *). A relation R( *) is "meta estimable" if it can be decomposed into terms of the form: Q k = P(V k do( k ), k ) such that each Q k is transportable from D k. AHEAICAL REUL #3: (ransportability and meta-transportability are solved) Nonparametric transportability of experimental results from multiple environments can be decided in polynomial time, provided commonalities and differences are encoded in selection diagrams. hen transportability is feasible, the transport formula can be derived in polynomial time, which specifies the information needed to be extracted from each environment to synthesize a consistent estimate for the target environment. he algorithm is complete. 24

25 DEERINING CAUE OF EFFEC A COUNERFACUAL VICOR our Honor! y client (r. A) died BECAUE he used that drug. Court to decide if it is ORE PROBABLE HAN NO that A would be alive BU FOR the drug! PN = P(? A is dead, took the drug) > 0.50 HE ARIBUION PROBLE Definition: 1. hat 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 PN(x, y) = P( x' = y' x, y) HE ARIBUION PROBLE Definition: 1. hat 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. Identification: 2. Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined. 25

26 ARIBUION AHEAIED (ian and Pearl, 2000) Bounds given combined nonexperimental and experimental data (P(y,x), P(y x ), for all y and x) max PN = 0 P(y) P(y x' ) P(x, y) 1 P(y' x' ) PN min P(x, y) Identifiability under monotonicity (Combined data) P(y x) P(y x') + P(y x') P(y x' ) P(y x) P(x, y) CAN FREQUENC DAA DECIDE LEGAL REPONIBILI? Experimental Nonexperimental do(x) do(x ) x x Deaths (y) urvivals (y ) ,000 1,000 1,000 1,000 Nonexperimental data: drug usage predicts longer life Experimental data: drug has negligible effect on survival Plaintiff: r. 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 Δ = P( x' = y' x, y) > 0.50 OLUION O HE ARIBUION PROBLE IH PROBABILI ONE 1 P(y x x,y) 1 Combined data tell more that each study alone 26

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