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CMSC 471CMSC 471Fall 2002Fall 2002

Class #19 – Monday, November 4

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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

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Bayesian Reasoning /Bayesian Reasoning /Bayesian NetworksBayesian Networks

Chapters 14, 15.1-15.2

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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)

aba b

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Inference from the joint: Examplealarm ¬alarm

earthquake ¬earthquake earthquake ¬earthquake

burglary .001 .008 .0001 .0009

¬burglary .01 .09 .001 .79

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/(.009+.1) = 9.173 (i.e., P(alarm) = 1/α = .109 – quizlet: how can you verify this?)

P(burglary | alarm) = .009 * 9.173 = .08255

P(¬burglary | alarm) = .1 * 9.173 = .9173

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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

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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

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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)).

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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):

onsanifestatievidence/m

hypotheses

1 mj

i

EEE

H

)(/)|()()|( jijiji EPHEPHPEHP

)( iHP

)|( ij HEP

)|( ij HEP

)|( ji EHP

)( iHP

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Simple Bayesian diagnostic reasoning

• Knowledge base:– Evidence / manifestations: E1, … Em

– Hypotheses / disorders: H1, … Hn

• Ej and Hi are binary; hypotheses are mutually exclusive (non-overlapping) 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)

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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) l

j=1 P(Ej | Hi)

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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 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 l

j=1 P(Ej | H1 H2) P(H1) P(H2)

• How do we compute P(Ej | H1 H2) ??

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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’ 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!

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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:

iiii xxP of nodesparent all ofset theis where)|(

)()|( so , iiii xPxP

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Example BN

a

b c

d e

P(C|A) = 0.2 P(C|~A) = 0.005P(B|A) = 0.3

P(B|~A) = 0.001

P(A) = 0.001

P(D|B,C) = 0.1 P(D|B,~C) = 0.01P(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

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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

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• 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 onlythrough 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

ix

)|(),...,( 11 iinin xPxxP

)|(),|( iiii xPqxP

ix i ix

i q

ix i

Independence and chaining

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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

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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) = Σ πi P(xi | πi) P(πi)

• In this equation, P(xi | πi) is given in the CPT, but computing P(πi) is complicated

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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

a

b c

d e

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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)

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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?

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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