bayes-nets.c19

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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, 15.1-15.2



4

Why probabilities anyway?

• Kolmogorov showed that three simple axioms lead to therules of probability theory

– De Finetti, Cox, and Carnap have also provided compellingarguments for these axioms

1. All probabilities are between 0 and 1:

• 0 <= P(a) <= 1

2. Valid propositions (tautologies) have probability 1, andunsatisfiable 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



5

Inference from the joint: Example

alarm ¬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



6

Independence

• When two sets of propositions do not affect each others’probabilities, we call them independent, and can easilycompute 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 beindependent 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 affectwhether we are burglarized

– Once we’re burglarized, light level doesn’t affect whether the alarmgoes off

• We need a more complex notion of independence, andmethods for reasoning about these kinds of relationships



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 jointprobability distribution



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)).



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 m j

i

E E E

H

)( / )|()()|( ji ji ji E P H E P H P E H P

)(i

H P

)|( i j H E P

)|( i j H E P

)|( ji E H P

)(i

H P



Simple Bayesian diagnostic reasoning

• Knowledge base:

– Evidence / manifestations: E1, … Em

– Hypotheses / disorders: H1, … Hn

• E j and Hi are binary; hypotheses are mutually exclusive (non-

overlapping) and exhaustive (cover all possible cases)

– Conditional probabilities: P(E j | 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)



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) = l j=1 P(E j | 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(E j | Hi)



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(E j | H1 H2) P(H1) P(H2)

• How do we compute P(E j | H1 H2) ??



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!



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 x x P of nodesparentallof settheiswhere)|(

)()|(so, iiii x P x P



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

i x

)|(),...,( 11 ii

n

i n x P x x P

)|(),|( iiii x Pq x P

i xi

i x

i

q

i x

i

Independence and chaining



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 priorprobability 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



20

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” andreduce 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(Xi | E=e)

– E.g., P(NoGas | Gauge=empty, Lights=on, Starts=false)

• Conjunctive queries:

– P(Xi, X j | E=e) = P(Xi | e=e) P(X j | 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?



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

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