1. COMMENTS ON By Judea Pearl (UCLA)

2. notation 1990’s Artificial Intelligence Hoover

3. Hoover slide From Hoover (2004) “Lost Causes”

4. notation 1990’s Artificial Intelligence

5. additional? ^ more? ^ Already unified

6. additional? ^ ^ Already permitted Commendable! (Statistical) P. 215

8. additional? ^ ^ Already permitted Commendable! (Statistical)

9. WHITE & CHALAK PICTURE OF UNIFICATION 1950 - 2005 SEM Neyman-Rubin DAGs Settable System  2006 MY PICTURE OF UNIFICATION 1920 - 1990 Informal SEM Neyman-Rubin Informal Diagrams Formal SEM  1990 - 20002006 (W&C) Complete Neyman-Rubin Graphs DAGs     Multi-agent extension

11. CAUSAL ANALYSIS WITHOUT TEARS

12. 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)

13. THE CAUSAL INFERENCE PARADIGM Data Inference Q(M) (Aspects of M) Data Generating Model Some Q(M) cannot be inferred from P. e.g., Infer whether customers who bought product A would still buy A if we were to double the price. Joint Distribution

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

15. Z Y X INPUTOUTPUT FAMILIAR CAUSAL MODEL ORACLE FOR MANIPILATION

16. WHY CAUSALITY NEEDS SPECIAL MATHEMATICS Y = 2X X = 1 Y = 2 Process information Had X been 3, Y would be 6. If we raise X to 3, Y would be 6. Must “wipe out” X = 1. Static information SEM Equations are Non-algebraic:

17. CAUSAL MODELS AND CAUSAL DIAGRAMS Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U

18. CAUSAL MODELS AND CAUSAL DIAGRAMS Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U U1U1 U2U2 IW Q P PA Q

19. Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U (iv)M x =  U,V,F x , X  V, x  X where F x = {f i : V i  X }  {X = x} (Replace all functions f i corresponding to X with the constant functions X=x) CAUSAL MODELS AND MUTILATION

20. CAUSAL MODELS AND MUTILATION Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U U1U1 U2U2 IW Q P (iv) (attributes)

21. CAUSAL MODELS AND MUTILATION Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U (iv) U1U1 U2U2 IW Q P P = p 0 MpMp (attributes)

22. Definition: A causal model is a 3-tuple M =  V,U,F  with a mutilation operator do(x): M  M x where: (i)V = {V 1 …,V n } endogenous variables, (ii)U = {U 1,…,U m } background variables (iii)F = set of n functions, f i : V \ V i  U  V i v i = f i (pa i,u i ) PA i  V \ V i U i  U (iv)M x =  U,V,F x , X  V, x  X where F x = {f i : V i  X }  {X = x} (Replace all functions f i 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. PROBABILISTIC CAUSAL MODELS

23. 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) and U=u, is equal to y. Joint probabilities of counterfactuals:

24. GRAPHICAL – COUNTERFACTUALS SYMBIOSIS Every causal model implies constraints on counterfactuals e.g., consistent, and readable from the graph. Every theorem in SEM is a theorem in N-R, and conversely.

25. GRAPHICAL TEST OF IDENTIFICATION The causal effect of X on Y, 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

26. RULES OF CAUSAL CALCULUS Rule 1: Ignoring observations P(y | do{x}, z, w) = P(y | do{x}, w) Rule 2: Action/observation exchange P(y | do{x}, do{z}, w) = P(y | do{x},z,w) Rule 3: Ignoring actions P(y | do{x}, do{z}, w) = P(y | do{x}, w)

27. RECENT RESULTS ON IDENTIFICATION Theorem (Tian 2002): We can identify P(v | do{x}) (x a singleton) if and only if there is no child Z of X connected to X by a bi-directed path. X ZZ Z k 1

28. do-calculus is complete A complete graphical criterion available for identifying causal effects of any set on any set References: Shpitser and Pearl 2006 (AAAI, UAI) RECENT RESULTS ON IDENTIFICATION (Cont.)

29. CONCLUSIONS Structural-model semantics enriched with logic + graphs leads to formal interpretation and practical assessments of wide variety of (if not all) causal and counterfactual relationships. causal effects, responsibility, direct and indirect effects Multi-agent systems? e.g.,

30. MULTI-AGENT GRAPHS Agent 1Agent 2 Y1Y1 Z1Z1 X1X1 u x1x1 u y1y1 u z1z1 Y2Y2 Z2Z2 X2X2 u x2x2 u y2y2 u z2z2

31. WHITE & CHALAK PICTURE OF UNIFICATION 1950 - 2005 SEM Neyman-Rubin DAGs Settable System  2006 MY PICTURE OF UNIFICATION 1920 - 1990 Informal SEM Neyman-Rubin Informal Diagrams Formal SEM  1990 - 20002006 (W&C) Complete Neyman-Rubin Graphs DAGs     Multi-agent extension