Bayes Net Perspectives on Causation and Causal Inference

1. Bayes Net Perspectives on Causation and Causal Inference Peter Spirtes

2. Example Problems • Genetic regulatory networks • Yeast – ~5000 genes, ~2,500,000 potential edges A gene regulatory network in mouse embryonic stem cells http://www.pnas.org/content/104/42/16438/F3.expansion.html

3. Causal Models → Predictions • Probabilistic – Among the cells that have active Oct4 what percentage have active Rcor2? • Causal – If I experimentally set a cell to have active Oct4, what percentage will have active Rcor2?

4. Causal Models → Predictions • Counterfactual – Among the cells that did not have active Oct4 at t-1, what percentage would have active Rcor2 if I had experimentally set a cell to have active Oct4 at t-1?

5. Data → Causal Models • Large number of variables • Small observed sample size • Overlapping variables • Small number of experiments • Feedback • Hidden common causes • Selection bias • Many kinds of entities causally interacting

6. Outline • Bayesian Networks • Search • Limitations and Extensions of Bayesian Networks • Dynamic • Relational • Cycles • Counterfactual

7. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Directed Acyclic Graph (DAG) SES SEX PE CP IQ SES – Socioeconomic Status PE – Parental Encouragement CP – College Plans IQ – Intelligence Quotient SEX– Sex • The vertices are random variables. • All edges are directed. • There are no directed cycles.

8. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Population SES SEX PE CP IQ SES SEX PE CP IQ SES SEX PE CP IQ Independent, identically distributed

9. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual P Factoring According to G SES SEX PE CP IQ • P(SES,SEX,PE,IQ,CP) = P(SEX)P(SES)P(IQ|SES) P(PE|SES,SEX,IQ) P(CP|PE,SES,IQ) • If • then P factors according to G • G represents all of the distributions that factor according to G

10. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Conditional Independence • X is independent of Y conditional on Z (denoted IP(X,Y|Z)) iff P(X|Y,Z) = P(X|Z). • IP(CP,SEX|{SES,IQ,PE}) iff P(CP|{SES,IQ,PE,SEX}) = P(CP|{SES,IQ,PE})

11. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Graphical Entailment SES SEX PE CP IQ • If for every P that factors according to G, IP(X,Y|Z) holds, then GentailsI(X,Y|Z). • Examples: G entails • I(IQ,SEX|∅) • I(IQ,SEX|SES) • Can read entailments off of graph through d-separation

12. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual D-separation and D-connection SES SEX PE CP IQ • X d-separated from Y conditional on Z in G iff G entails X independent of Y conditional on Z • D-separation between X and Y conditional on Z holds when certain kinds of paths do notexist between X and Y • D-connection (the negation of d-separation) between X and Y conditional on Z holds when certain kinds of paths do exist between X and Y

13. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Definition of D-connection SES SEX PE CP IQ • A node X is active on a path UconditionalonZ iff • X is a collider (→ X ←) and there is a directed path from X to a member of Z or X is in Z; or • X is not a collider and X is not in Z. • SES → IQ → PE ← SEX is a path U. • PE is active on U conditional on {CP, IQ}. • IQ is inactive on U conditional on {CP, IQ}.

14. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Definition of D-connection SES SEX PE CP IQ • A path U is active conditional onZ iff every vertex on U is active relative to Z. • X is d-connectedto Y conditional onZ iff there is an active path between X and Y conditional on Z. • SES → IQ → PE ← SEX is inactive conditional on{CP, IQ}. • SES is d-connected to SEX conditional on {CP, IQ} because SES → PE ← SEX is active conditional on {CP, IQ}

15. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual If I is Not Entailed by G • If conditional independence relation I is not entailed by G, then I may hold in some (but not every) distribution P that factors according to G. SES SEX PE CP IQ • Example: There are P and P’ that factor according to G such that ~IP(SES,CP|∅) and IP’(SES,CP|∅). P’ is said to be unfaithful to G.

16. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Manipulations • An ideal manipulationassigns a density to a set X of properties (random variables) as a function of the values of a set Z of properties (random variables) • Directly affects only the variables in X • Successful • Example – randomized experiment

17. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Manipulations and Causal Graph SES SEX PE CP IQ • There is an edge SES → CP in Gbecause there are two ways of manipulating {SES,SEX,IQ,PE} that differ only in the value they assign to SES that changes the probability of CP. Stable Unit Treatment Value Assumption

18. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Causal Sufficiency SES SEX PE CP IQ • A set S of variables is causally sufficient if there are no variables not in S that are direct causes of more than one variable in S. • S = {SES,IQ} is causally sufficient. • S = {SES,PE,CP} is not causally sufficient.

19. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Causal Markov Assumption SES SEX PE CP IQ • In a population Pop with distribution P and causal graph G, if V is causally sufficient, P(V) factorsaccording to G. • P(SES,SEX,PE,IQ,CP) = P(SEX)P(SES)P(IQ|SES) P(PE|SES,SES,IQ) P(CP|PE,SES,IQ)

20. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Representation of Manipulation SES SEX PE CP IQ P(SES,SEX,PE=1,IQ,CP||PE=1) = P(SEX)P(SES)P(IQ|SES) * 1 * P(CP|PE,SES,IQ) = P(SES,SEX,PE=1,IQ,CP)/P(PE|SEX,SES,IQ)

21. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual FCI Algorithm • Looks for set of DAGs (possibly with latent variables and selection bias) that entail all and only the conditional independence relations that hold in the data according to statistical tests.

22. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Markov Equivalence • Two DAGs G1 and G2 are Markov equivalent when they contain the same variables, and for all disjoint X, Y, Z, X is entailed to be independent from Y conditional on Z in G1 if and only if X is entailed to be independent from Y conditional on Z in G2

23. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Markov Equivalence Class SES SEX PE CP IQ SES SEX PE CP IQ DAG G’ DAG G

24. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Causal Faithfulness Assumption SES SEX PE CP IQ • In a population Pop with causal graph G and distribution P(V), if V is causally sufficient, IP(X,Y|Z) only if G entails I(X,Y|Z). • ~IP(SES,CP|∅) because I(SES,CP|∅)is not entailed by G • +…

25. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Causal Faithfulness Assumption SES SEX PE CP IQ • Causal Faithfulness is too strong because • can prove consistency with assumptions about fewer conditional independencies • is unlikely to hold, especially when there are many variables. • Causal Faithfulness is too weak because it is not sufficient to prove uniform consistency (put error bounds at finite sample sizes.)

26. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Good Features of FCI Algorithm • Is pointwise consistent: As sample size → ∞, P(error in output pattern) → 0. • Can be applied to distributions where tests of conditional independence are known • Can be applied to hidden variable models (and selection bias models)

27. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Bad Features of FCI Algorithm • There is no reliable way to set error bounds on the pattern without making stronger assumptions. • Can only get set of Markov equivalent DAGs, not a single DAG • Doesn’t allow for comparing how much better one model is than another • Need to assume some version of Causal Faithfulness Assumption

28. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Non Independence Constraints • Depending on the parametric family, a DAG can entail constraints that are not conditional independence constraints • Assuming linearity and non-Gaussian error terms, if a distribution is compatible with X → Y it is not compatible with X ← Y, even though they are Markov equivalent.

29. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Score-Based Search Strategy • Assign score to graph and sample based on • maximum likelihood of data given graph • simplicity of model • Do search over graph space for highest score

30. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Advantages of Score-Based Search Strategy • Get more information about graph • Additive noise models, unique DAG • Doesn’t rely on binary decisions • Local mistakes don’t propagate

31. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Disadvantages of Score-Based Search Strategy • Often slower to calculate or not known how to calculate exactly if include • unmeasured variables • selection bias • unusual distributions • Search over graph space is often heuristic

32. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Dynamic Bayes Nets • If measure same variable at different times, then the samples from the variable are not i.i.d. • Solution: index each variable by time (time series)

33. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Dynamic Bayes Nets • Make a template for the causal structure that can be filled in with actual times Xt-2Xt-1Xt Yt-2Yt-1Yt • Continuous time or differential equations? • Continuous time or differential equations?

34. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual parent-of parent-of parent-of Population SES SEX PE CP IQ SES SEX PE CP IQ SES SEX PE CP IQ

35. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual parent-of parent-of parent-of Population SES SEX PE CP IQ • Not i.i.d. distribution • Violations of SUTVA • Causal relations between relations (e.g. sibling causes rivalry)

36. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Extended Manipulation Specification • A manipulation assigns a density to • a set of properties or relations • at a set of times (measurable set of times T) • for a set of units • as a function of the values of • a set of properties of relations • at a set of times (measurable set of times T) • for a set of units

37. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual parent-of parent-of Extended Factorization Assumption Alice&Jim SES SEX PE CP IQ Sue Bob P([Alice&Jim.SES, Sue.SEX,Sue.PE, Sue.IQ, Sue.CP, Alice&Jim.SES, Bob.SEX,Bob.PE, Bob.IQ, Bob.CP) =

38. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual Extended Factorization Assumption P(Sue.SEX) P(Alice&Jim.SES)P(Sue.IQ|Alice&Jim .SES) P(Sue.PE|Alice&Jim.SES,Sue.SEX, Sue.IQ) P(Sue.CP|Sue.PE, Alice&Jim.SES, Sue.IQ) P(Bob.SEX) P(Alice&Jim.SES) P(Bob.IQ|Alice&Jim.SES) P(Bob.PE|Alice&Jim.SES, Bob.SEX, Bob.IQ) P(Bob.CP|Bob.PE, Alice&Jim.SES, Bob.IQ)

39. Bayesian Networks • Search • Limitations and Extensions • Dynamic • Relational • Cycles • Counterfactual 3 Interpretation of Cycles: PE ⇆ CP SES SEX PE CP IQ • Equilibrium values of PE and CP cause each other. • Average of values of PE and CP while reaching equilibrium influence each other. • Mixture of PE→ CP and CP→ PE