1 CMSC 471 Fall 2002 Class #19 – Monday, November 4

2 Today’s class (Probability theory) Bayesian inference –From the joint distribution –Using independence/factoring –From sources of evidence Bayesian networks –Network structure –Conditional probability tables –Conditional independence –Inference in Bayesian networks

3 Bayesian Reasoning / Bayesian Networks Chapters 14,

4 Why probabilities anyway? Kolmogorov showed that three simple axioms lead to the rules of probability theory –De Finetti, Cox, and Carnap have also provided compelling arguments for these axioms 1.All probabilities are between 0 and 1: 0 <= P(a) <= 1 2.Valid propositions (tautologies) have probability 1, and unsatisfiable propositions have probability 0: P(true) = 1 ; P(false) = 0 3.The probability of a disjunction is given by: P(a  b) = P(a) + P(b) – P(a  b) abab a b

5 Inference from the joint: Example alarm¬alarm earthquake¬earthquakeearthquake¬earthquake burglary ¬burglary P(Burglary | alarm) = α P(Burglary, alarm) = α [P(Burglary, alarm, earthquake) + P(Burglary, alarm, ¬earthquake) = α [ (.001,.01) + (.008,.09) ] = α [ (.009,.1) ] Since P(burglary | alarm) + P(¬burglary | alarm) = 1, α = 1/( ) = (i.e., P(alarm) = 1/α =.109 – quizlet: how can you verify this?) P(burglary | alarm) =.009 * = P(¬burglary | alarm) =.1 * =.9173

6 Independence When two sets of propositions do not affect each others’ probabilities, we call them independent, and can easily compute their joint and conditional probability: –Independent (A, B) → P(A  B) = P(A) P(B), P(A | B) = P(A) For example, {moon-phase, light-level} might be independent of {burglary, alarm, earthquake} –Then again, it might not: Burglars might be more likely to burglarize houses when there’s a new moon (and hence little light) –But if we know the light level, the moon phase doesn’t affect whether we are burglarized –Once we’re burglarized, light level doesn’t affect whether the alarm goes off We need a more complex notion of independence, and methods for reasoning about these kinds of relationships

7 Conditional independence 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) 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

8 Bayes’ rule Bayes rule is derived from the product rule: –P(Y | X) = P(X | Y) P(Y) / P(X) Often useful for diagnosis: –If X are (observed) effects and Y are (hidden) causes, –We may have a model for how causes lead to effects (P(X | Y)) –We may also have prior beliefs (based on experience) about the frequency of occurrence of effects (P(Y)) –Which allows us to reason abductively from effects to causes (P(Y | X)).

9 Bayesian inference In the setting of diagnostic/evidential reasoning –Know prior probability of hypothesis conditional probability –Want to compute the posterior probability Bayes’ theorem (formula 1):

10 Simple Bayesian diagnostic reasoning Knowledge base: –Evidence / manifestations:E 1, … E m –Hypotheses / disorders:H 1, … H n E j and H i are binary; hypotheses are mutually exclusive (non- overlapping) and exhaustive (cover all possible cases) –Conditional probabilities:P(E j | H i ), i = 1, … n; j = 1, … m Cases (evidence for a particular instance): E 1, …, E l Goal: Find the hypothesis H i with the highest posterior –Max i P(H i | E 1, …, E l )

11 Bayesian diagnostic reasoning II Bayes’ rule says that –P(H i | E 1, …, E l ) = P(E 1, …, E l | H i ) P(H i ) / P(E 1, …, E l ) Assume each piece of evidence E i is conditionally independent of the others, given a hypothesis H i, then: –P(E 1, …, E l | H i ) =  l j=1 P(E j | H i ) If we only care about relative probabilities for the Hi, then we have: –P(H i | E 1, …, E l ) = α P(H i )  l j=1 P(E j | H i )

12 Limitations of simple Bayesian inference Cannot easily handle multi-fault situation, nor cases where intermediate (hidden) causes exist: –Disease D causes syndrome S, which causes correlated manifestations M 1 and M 2 Consider a composite hypothesis H 1  H 2, where H 1 and H 2 are independent. What is the relative posterior? –P(H 1  H 2 | E 1, …, E l ) = α P(E 1, …, E l | H 1  H 2 ) P(H 1  H 2 ) = α P(E 1, …, E l | H 1  H 2 ) P(H 1 ) P(H 2 ) = α  l j=1 P(E j | H 1  H 2 ) P(H 1 ) P(H 2 ) How do we compute P(E j | H 1  H 2 ) ??

13 Limitations of simple Bayesian inference II Assume H1 and H2 are independent, given E1, …, El? –P(H 1  H 2 | E 1, …, E l ) = P(H 1 | E 1, …, E l ) P(H 2 | E 1, …, E l ) This is a very unreasonable assumption –Earthquake and Burglar are independent, but not given Alarm: P(burglar | alarm, earthquake) << P(burglar | alarm) Another limitation is that simple application of Bayes’ rule doesn’t allow us to handle causal chaining: –A: year’s weather; B: cotton production; C: next year’s cotton price –A influences C indirectly: A→ B → C –P(C | B, A) = P(C | B) Need a richer representation to model interacting hypotheses, conditional independence, and causal chaining Next time: conditional independence and Bayesian networks!

14 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:

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

16 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

17 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 Independence and chaining

18 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) a b c d e

19 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(x i ) If x i is a root, then P(x i ) is given directly in the BN (CPT at X i ) Otherwise, –P(x i ) = Σ πi P(x i | π i ) P(π i ) In this equation, P(x i | π i ) is given in the CPT, but computing P(π i ) is complicated

20 Computing π i : 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 a b c d e

21 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)

22 Inference tasks Simple queries: Computer posterior marginal P(X i | E=e) –E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false) Conjunctive queries: –P(X i, X j | E=e) = P(X i | e=e) P(X j | X i, 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?

23 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