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Conditional Independence and Bayesian Networks

This class discusses conditional independence, Bayesian networks, network structure, conditional probability tables, and inference in Bayesian networks. Topics include topological semantics, independence and chaining, direct inference, representational extensions, and inference tasks.

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Conditional Independence and Bayesian Networks

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  1. CMSC 671Fall 2001 Class #20 – Thursday, November 8

  2. Today’s class • Conditional independence • Bayesian networks • Network structure • Conditional probability tables • Conditional independence • Inference in Bayesian networks

  3. Conditional Independence and Bayesian Networks Chapters 13/14 (2/e)

  4. Conditional independence • Last time we talked about absolute independence: • A and B are independent if P(A  B) = P(A) P(B); equivalently, P(A) = P(A | B) and P(B) = P(B | A) • A and B are conditionally independent given C if • P(A  B | C) = P(A | C) P(B | C) • This lets us decompose the joint distribution: • P(A  B  C) = P(A | C) P(B | C) P(C) • In the example from last time, Moon-Phase and Burglary are conditionally independent given Light-Level • Conditional independence is weaker than absolute independence, but still useful in decomposing the full joint probability distribution

  5. Bayesian Belief Networks (BNs) • Definition: BN = (DAG, CPD) • DAG: directed acyclic graph (BN’s structure) • Nodes: random variables (typically binary or discrete, but methods also exist to handle continuous variables) • Arcs: indicate probabilistic dependencies between nodes (lack of link signifies conditional independence) • CPD: conditional probability distribution (BN’s parameters) • Conditional probabilities at each node, usually stored as a table (conditional probability table, or CPT) • Root nodes are a special case – no parents, so just use priors in CPD:

  6. a b c d e Example BN P(A) = 0.001 P(C|A) = 0.2 P(C|~A) = 0.005 P(B|A) = 0.3 P(B|~A) = 0.001 P(D|B,C) = 0.1 P(D|B,~C) = 0.01 P(D|~B,C) = 0.01 P(D|~B,~C) = 0.00001 P(E|C) = 0.4 P(E|~C) = 0.002 Note that we only specify P(A) etc., not P(¬A), since they have to add to one

  7. Topological semantics • A node is conditionally independent of its non-descendants given its parents • A node is conditionally independent of all other nodes in the network given its parents, children, and children’s parents (also known as its Markov blanket) • The method called d-separation can be applied to decide whether a set of nodes X is independent of another set Y, given a third set Z

  8. Independence and chaining • Independence assumption where q is any set of variables (nodes) other than and its successors • blocks influence of other nodes on and its successors (q influences only through variables in ) • With this assumption, the complete joint probability distribution of all variables in the network can be represented by (recovered from) local CPD by chaining these CPD q

  9. a b c d e Chaining: Example Computing the joint probability for all variables is easy: P(a, b, c, d, e) = P(e | a, b, c, d) P(a, b, c, d) by Bayes’ theorem = P(e | c) P(a, b, c, d) by indep. assumption = P(e | c) P(d | a, b, c) P(a, b, c) = P(e | c) P(d | b, c) P(c | a, b) P(a, b) = P(e | c) P(d | b, c) P(c | a) P(b | a) P(a)

  10. Direct inference with BNs • Now suppose we just want the probability for one variable • Belief update method • Original belief (no variables are instantiated): Use prior probability p(xi) • If xi is a root, then P(xi) is given directly in the BN (CPT at Xi) • Otherwise, • P(xi) = Σπi P(xi | πi) P(πi) • In this equation, P(xi | πi) is given in the CPT, but computing P(πi) is complicated

  11. a b c d e Computing πi: Example • P (d) = P(d | b, c) P(b, c) • P(b, c) = P(a, b, c) + P(¬a, b, c) (marginalizing)= P(b | a, c) p (a, c) + p(b | ¬a, c) p(¬a, c) (product rule)= P(b | a) P(c | a) P(a) + P(b | ¬a) P(c | ¬a) P(¬a) • If some variables are instantiated, can “plug that in” and reduce amount of marginalization • Still have to marginalize over all values of uninstantiated parents – not computationally feasible with large networks

  12. Representational extensions • Compactly representing CPTs • Noisy-OR • Noisy-MAX • Adding continuous variables • Discretization • Use density functions (usually mixtures of Gaussians) to build hybrid Bayesian networks (with discrete and continuous variables)

  13. Inference tasks • Simple queries: Computer posterior marginal P(Xi | E=e) • E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false) • Conjunctive queries: • P(Xi, Xj | E=e) = P(Xi | e=e) P(Xj | Xi, E=e) • Optimal decisions:Decision networks include utility information; probabilistic inference is required to find P(outcome | action, evidence) • Value of information: Which evidence should we seek next? • Sensitivity analysis:Which probability values are most critical? • Explanation: Why do I need a new starter motor?

  14. Approaches to inference • Exact inference • Enumeration • Variable elimination • Clustering / join tree algorithms • Approximate inference • Stochastic simulation / sampling methods • Markov chain Monte Carlo methods • Genetic algorithms • Neural networks • Simulated annealing • Mean field theory

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