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Bayesian Reasoning Chapter 13 1 Today’s class • Probability theory • Bayesian inference – From the joint distribution – Using independence/factoring – From sources of evidence 2 Sources of uncertainty • Uncertain inputs – Missing data – Noisy data • Uncertain knowledge – Multiple causes lead to multiple effects – Incomplete enumeration of conditions or effects – Incomplete knowledge of causality in the domain – Probabilistic/stochastic effects • Uncertain outputs – Abduction and induction are inherently uncertain – Default reasoning, even in deductive fashion, is uncertain – Incomplete deductive inference may be uncertain Probabilistic reasoning only gives probabilistic results (summarizes uncertainty from various sources) 3 Decision making with uncertainty Rational behavior: – For each possible action, identify the possible outcomes – Compute the probability of each outcome – Compute the utility of each outcome – Compute the probability-weighted (expected) utility over possible outcomes for each action – Select the action with the highest expected utility (principle of Maximum Expected Utility) 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 ab b 5 Probability theory • Random variables – Domain • Alarm, Burglary, Earthquake – Boolean (like these), discrete, continuous • Atomic event: complete specification of state • Alarm=True Burglary=True Earthquake=False alarm burglary ¬earthquake • Prior probability: degree of belief without any other evidence • Joint probability: matrix of combined probabilities of a set of variables • P(Burglary) = .1 • P(Alarm, Burglary) = alarm ¬alarm burglary .09 .01 ¬burglary .1 .8 6 Probability theory (cont.) • Conditional probability: probability of effect given causes • Computing conditional probs: – P(a | b) = P(a b) / P(b) – P(b): normalizing constant • Product rule: – P(a b) = P(a | b) P(b) • Marginalizing: – P(B) = ΣaP(B, a) – P(B) = ΣaP(B | a) P(a) (conditioning) • P(burglary | alarm) = .47 P(alarm | burglary) = .9 • P(burglary | alarm) = P(burglary alarm) / P(alarm) = .09 / .19 = .47 • P(burglary alarm) = P(burglary | alarm) P(alarm) = .47 * .19 = .09 • P(alarm) = P(alarm burglary) + P(alarm ¬burglary) = .09+.1 = .19 7 Example: Inference from the joint alarm ¬alarm earthquake ¬earthquake earthquake ¬earthquake burglary .01 .08 .001 .009 ¬burglary .01 .09 .01 .79 P(Burglary | alarm) = α P(Burglary, alarm) = α [P(Burglary, alarm, earthquake) + P(Burglary, alarm, ¬earthquake) = α [ (.01, .01) + (.08, .09) ] = α [ (.09, .1) ] Since P(burglary | alarm) + P(¬burglary | alarm) = 1, α = 1/(.09+.1) = 5.26 (i.e., P(alarm) = 1/α = .19 – quizlet: how can you verify this?) P(burglary | alarm) = .09 * 5.26 = .474 P(¬burglary | alarm) = .1 * 5.26 = .526 8 Exercise: Inference from the joint smart smart p(smart study prep) study study study study prepared .432 .16 .084 .008 prepared .048 .16 .036 .072 • Queries: – What is the prior probability of smart? – What is the prior probability of study? – What is the conditional probability of prepared, given study and smart? • Save these answers for next time! 9 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 10 Exercise: Independence smart smart p(smart study prep) study study study study prepared .432 .16 .084 .008 prepared .048 .16 .036 .072 • Queries: – Is smart independent of study? – Is prepared independent of study? 11 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 12 Exercise: Conditional independence smart smart p(smart study prep) study study study study prepared .432 .16 .084 .008 prepared .048 .16 .036 .072 • Queries: – Is smart conditionally independent of prepared, given study? – Is study conditionally independent of prepared, given smart? 13 Bayes’s rule • Bayes’s 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)). 14 Bayesian inference • In the setting of diagnostic/evidential reasoning H i P(Hi ) hypotheses P(E j | Hi ) E1 Ej Em evidence/m anifestati ons – Know prior probability of hypothesis conditional probability – Want to compute the posterior probability P(Hi ) P(E j | Hi ) P(Hi | E j ) • Bayes’s theorem (formula 1): P(Hi | E j ) P(Hi )P(E j | Hi ) / P(E j ) 15 Simple Bayesian diagnostic reasoning • Knowledge base: – Evidence / manifestations: – Hypotheses / disorders: E1, … Em H1, … H n • Ej and Hi are binary; hypotheses are mutually exclusive (nonoverlapping) and exhaustive (cover all possible cases) – Conditional probabilities: P(Ej | Hi), i = 1, … n; j = 1, … m • Cases (evidence for a particular instance): E1, …, El • Goal: Find the hypothesis Hi with the highest posterior – Maxi P(Hi | E1, …, El) 16 Bayesian diagnostic reasoning II • Bayes’ rule says that – P(Hi | E1, …, El) = P(E1, …, El | Hi) P(Hi) / P(E1, …, El) • Assume each piece of evidence Ei is conditionally independent of the others, given a hypothesis Hi, then: – P(E1, …, El | Hi) = lj=1 P(Ej | Hi) • If we only care about relative probabilities for the Hi, then we have: – P(Hi | E1, …, El) = α P(Hi) lj=1 P(Ej | Hi) 17 Limitations of simple Bayesian inference • Cannot easily handle multi-fault situations, nor cases where intermediate (hidden) causes exist: – Disease D causes syndrome S, which causes correlated manifestations M1 and M2 • Consider a composite hypothesis H1 H2, where H1 and H2 are independent. What is the relative posterior? – P(H1 H2 | E1, …, El) = α P(E1, …, El | H1 H2) P(H1 H2) = α P(E1, …, El | H1 H2) P(H1) P(H2) = α lj=1 P(Ej | H1 H2) P(H1) P(H2) • How do we compute P(Ej | H1 H2) ?? 18 Limitations of simple Bayesian inference II • Assume H1 and H2 are independent, given E1, …, El? – P(H1 H2 | E1, …, El) = P(H1 | E1, …, El) P(H2 | E1, …, El) • 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’s rule doesn’t allow us to handle causal chaining: – A: this 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! 19