1. 1 Bayesian Networks and Causal Modelling Ann Nicholson School of Computer Science and Software Engineering Monash University

2. 2 Overview Introduction to Bayesian Networks (BNs) Summary of BN research projects Varieties of Causal intervention » PRICAI2004: K. Korb, L. Hope, A. Nicholson, K. Axnick Learning Causal Structure » CaMML software

3. 3 Probability theory for representing uncertainty Assigns a numerical degree of belief between 0 and 1 to facts » e.g. “it will rain today” is T/F. » P(“it will rain today”) = 0.2 prior probability (unconditional) Posterior probability (conditional) » P(“it wil rain today” | “rain is forecast”) = 0.8 Bayes’ Rule: P(H|E) = P(E|H) x P(H) P(E)

4. 4 Bayesian networks A Bayesian Network (BN) represents a probability distribution graphically (directed acyclic graphs) Nodes: random variables, » R: “it is raining”, discrete values T/F » T: temperature, cts or discrete variable » C: colour, discrete values {red,blue,green} Arcs indicate conditional dependencies between variables P(A,S,T) can be decomposed to P(A)P(S|A)P(T|A)

5. 5 Bayesian networks (cont.) There is a conditional probability distribution (CPD or CPT) associated with each node. – probability of each state given parent states “Jane has the flu” “Jane has a high temp” “Thermometer temp reading” X Flu Y Te Q Th Models causal relationship Models possible sensor error P(Flu=T) = 0.05 P(Te=High|Flu=T) = 0.4 P(Te=High|Flu=F) = 0.01 P(Th=High|Te=H) = 0.95 P(Th=High|Te=L) = 0.1

6. 6 BN inference Evidence: observation of specific state Task: compute the posterior probabilities for query node(s) given evidence. Th Y Flu Te Diagnostic inference ThFlu Y Te Predictive inference Intercausal inference Te FluTBFlu Mixed inference ThFlu Te

7. 7 Causal Networks Arcs follow the direction of causal process Causal Networks are always BNs Bayesian Networks aren't always causal

8. 8 Early BN-related projects DBNS for discrete monitoring (PhD, 1992) Approximate BN inference algorithms based on a mutual information measure for relevance ( with Nathalie Jitnah, 1996-1999) Plan recognition: DBNs for predicting users actions and goals in an adventure game ( with David Albrecht, Ingrid Zukerman, 1997-2000 ) DBNs for ambulation monitoring and fall diagnosis ( with biomedical engineering, 1996-2000 ) Bayesian Poker ( with Kevin Korb, 1996-2003 )

9. 9 Knowledge Engineering with BNs Seabreeze prediction: joint project with Bureau of Meteorology » Comparison of existing simple rule, expert elicited BN, and BNs from Tetrad-II and CaMML ITS for decimal misconceptions Methodology and tools to support knowledge engineering process » Matilda: visualisation of d-separation » Support for sensitivity analysis Written a textbook: » Bayesian Artificial Intelligence, Kevin B. Korb and Ann E. Nicholson, Chapman & Hall / CRC, 2004. www.csse.monash.edu.au/bai/book

10. 10 Current BN-related projects BNs for Epidemiology (with Kevin Korb, Charles Twardy) » ARC Discovery Grant, 2004 » Looking at Coronary Heart Disease data sets » Learning hybrid networks: cts and discrete variables. BNs for supporting meteorological forecasting process (DSS’2004) (with Ph. D student Tal Boneh, K. Korb, BoM) » Building domain ontology (in Protege) from expert elicitation » Automatically generating BN fragments » Case studies: Fog, hailstorms, rainfall. Ecological risk assessment » Goulburn Water, native fish abundance » Sydney Harbour Water Quality

11. 11 Other projects Autonomous aircraft monitoring and replanning ( with Ph.D. student Tim Wilkin, PRICAI2000, IAV2004 ) Dynamic non-uniform abstraction for approximate planning with MDPs (with Ph.D. student Jiri Baum)

12. 12 Observation and Intervention Inference from observations » Predictive reasoning (finding effects) » Diagnostic reasoning (finding causes) Inference with interventions » Predictive reasoning » Not diagnostic reasoning Causal reasoning shouldn't go against causality. Th Flu Te Diagnostic inference Th Flu Te Predictive inference

13. 13 Pearlian Determinism Pearl's reasons for determinism: » Determinism is intuitive » Counterfactuals and causal explanation only make sense with a deterministic interpretation » Any indeterministic model can be transformed into a deterministic model We see no reason for assuming determinism

14. 14 Defining Intervention I Arc cutting » More intuitive » Intervention node Intervention node » More general interventions » Much easier to implement » To simulate arc cutting: P(C|  c, I c )=1 Arc cutting isn’t general enough

15. 15 Defining Intervention II We define an intervention on model M as: M augmented with I c (M') where: 1. I c has the purpose of manipulating C 2. I c is exogenous (has no parents) in M' 3. I c directly causes (is a parent of) C To preserve the original network: P M' (C|  c,¬ I c ) = P M' (C|  c ) where  c are the original parents of C. We also define P * (C) as the intended distribution.

16. 16 Varieties of Intervention: Dependency The degree of dependency of the effect upon existing parents. An independent intervention cuts the child off from its other parents. Thus, P M' (C|  c, I c ) = P * (C) A dependent intervention allows any parent interaction.

17. 17 Varieties of Intervention : Indeterminism The degree of indeterminism of the effect. A deterministic intervention sets the child to one particular state. A stochastic intervention sets the child to a positive distribution. Dependency and Determinism characterize any intervention Pearlian interventions are independent and deterministic

18. 18 Varieties of Intervention : Effectiveness We've found the idea of effectiveness useful. If P * (C) is what's intended and r is the effectiveness, then P M' (C |  c, I c ) = r × P * (C) + (1-r) × P M' (C |  c ) This is a dependent intervention.

19. 19 Demo of Causal Intervention Software

20. 20 Summary of Causal Intervention A taxonomy of intervention types More realistic interventions (e.g., partial effectiveness) A GUI which handles some varieties of intervention » Pearlian » Partially effective » Extensible to deal with other types of interaction explicitly

21. 21 Learning Causal Structure This is the real problem; parameterizing models is relatively straightforward estimation problem. Size of the dag space is superexponential: » Number of possible orderings: n! » Times number of possible arcs: C n 2 » Minus number of possible cyclic graphs More exactly (Robinson, 1977): f(n) =  (-1) i+1 C n i 2 i(n-i) f(n-i) so for » n=3, f(n)=25 » n=5, f(n)=25,000 » n=10, f(n)  4.2x10 18

22. 22 Learning Causal Structure There are two basic methods: » Learning from conditional independencies (CI learning) » Learning using a scoring metric (Metric learning) CI learning (Verma and Pearl, 1991) » Suppose you have an Oracle who can answer yes or no to any question of the type: is X conditional independence Y given S? » Then you can learn the correct causal model, up to statistical equivalence (patterns).

23. 23 Statistical Equivalence Two causal models H 1 and H 2 are statistically equivalent iff they contain the same variables and joint samples over them provide no statistical grounds for preferring one over the other. Examples » All fully connected models are equivalent. » A  B  C and A  B  C. » A  B  D  C and A  B  D  C.

24. 24 Statistical Equivalence (cont.) (Verma and Pearl, 1991): Any two causal models over the same variables which have the same skeleton (undirected arcs) and the same directed v-structures are statistically equivalent. Chickering (1995): If H 1 and H 2 are statistically equivalent, then they have the same maximum likelihoods relative to any joint samples max P(e|H 1,  1 ) = max P(e|H 2,  2 ) where  i is a parameterization of H i

25. 25 Other approaches to structure learning TETRAD II: Spirtes, Glymour and Scheines (1993). Implemented in their PC algorithm » Doesn't handle well with weak links and small samples (demonstrated empirically in Dai, Korb, Wallace & Wu (1997)). Bayesian LBN: Cooper & Herskovits' K2 (1991, 1992) » Compute P(h i |e) by brute force, under the various assumptions which reduce the computation of P CH (h,e) to a polynomial time counting problem. » But the hypothesis space is exponential; they go for dramatic simplification by assuming we know the temporal ordering of the variables.

26. 26 Learning Variable Order Reliance upon a given variable order is a major drawback to K2 » And many other algorithms (Buntine 1991, Bouckert 1994, Suzuki 1996, Madigan & Raftery 1994) What's wrong with that? » We want autonomous AI (data mining). If experts can order the variables they can likely supply models. » Determining variable ordering is half the problem. If we know A comes before B, the only remaining issue is whether there is a link between the two. » The number of orderings consistent with dags is exponential (Brightwell & Winkler 1990; number complete). So iterating over all possible orderings will not scale up.

27. 27 Statistical Equivalence Learners Heckerman & Geiger (1995) advocate learning only up to statistical equivalence classes (a la TETRAD II). » Since observational data cannot distinguish btw equivalent models, there's no point trying to go further.  Madigan, Andersson, Perlman & Volinsky (1996) follow this advice, use uniform prior over equivalence classes.  Geiger and Heckerman (1994) define Bayesian metrics for linear and discrete equivalence classes of models (BGe and BDe)

28. 28 Statistical Equivalence Learners Wallace & Korb (1999): This is not right! These are causal models; they are distinguishable on experimental data. » Failure to collect some data is no reason to change prior probabilities. E.g., If your thermometer topped out at 35C, you wouldn't treat  35C and 34C as equally likely. Not all equivalence classes are created equal: { A  B  C, A  B  C, A  B  C } { A  B  C } Within classes some dags should have greater priors than others… E.g., » LightsOn  InOffice  LoggedOn v. » LightsOn  InOffice  LoggedOn

29. 29 Full Causal Learners So… a full causal learner is an algorithm that: 1. Learns causal connectedness. 2. Learns v-structures. Hence, learns equivalence classes. 3. Learns full variable order. Hence, learns full causal structure (order + connectedness). TETRAD II: 1, 2. Madigan et al.; Heckerman & Geiger (BGe, BDe): 1, 2. Cooper & Herskovits' K2: 1. Lam and Bacchus MDL: 1, 2 (partial), 3 (partial). Wallace, Neil, Korb MML: 1, 2, 3.

30. 30 CaMML Minimum Message Length (Wallace \& Boulton 1968) uses Shannon's measure of information: I(m) = - log P(m) Applied in reverse, we can compute P(h,e) from I(h,e). Given an efficient joint encoding method for the hypothesis & evidence space (i.e., satisfying Shannon's law), MML: Searches {h i } for that hypothesis h that minimizes I(h) + I(e|h). Applies a trade-off between » Model simplicity » Data fit Equivalent to that h that maximizes P(h)P(e|h) --- i.e., P(h|e).

31. 31 MML search algorithms MML metrics need to be combined with search. This has been done three ways: 1. Wallace, Korb, Dai (1996): greedy search (linear). » Brute force computation of linear extensions (small models only) 2. Neil and Korb (1999): genetic algorithms (linear). » Asymptotic estimator of linear extensions » GA chromosomes = causal models » Genetic operators manipulate them » Selection pressure is based on MML Wallace and Korb (1999): MML sampling (linear, discrete). » Stochastic sampling through space of totally ordered causal models » No counting of linear extensions required

32. 32 Empirical Results A weakness in this area --- and AI generally. » Papers based upon very small models, loose comparisons. » ALARM often used --- everything gets it to within 1 or 2 arcs. Neil and Korb (1999) compared CaMML and BGe (Heckerman & Geiger's Bayesian metric over equivalence classes), using identical GA search over linear models: » On KL distance and topological distance from the true model, CaMML and BGe performed nearly the same. » On test prediction accuracy on strict effect nodes (those with no children), CaMML clearly outperformed BGe.

33. 33 Extensions to original CaMML Allow specification of prior on arc » O’Donnell, Korb, Nicholson » Useful for combining expert and automated methods Learning local structure » Logit models (Neill, Wallace, Korb) » Hybrid networks - CPT or decision trees (O’Donnell, Allison, Korb, Hope) (Uses MCMC search)

34. 34 CaMML Information and executables available at www.datamining.monash.edu.au/software/camml Linear and Discrete versions Weka wrapper available