Download CS 471 - Bayesian Networks

CMSC 471
Fall 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
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–
–
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Network structure
Conditional probability tables
Conditional independence
Inference in Bayesian networks
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Bayesian Reasoning /
Bayesian 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)
a
ab
b
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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
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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
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’ theorem (formula 1):
P(Hi | E j )  P(Hi )P(E j | Hi ) / P(E j )
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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)
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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) lj=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 lj=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)
P ( xi |  i ) where  i is the set of all parent nodes of xi
– Root nodes are a special case – no parents, so just use priors
in CPD:
 i  , so P ( xi |  i )  P ( xi )
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Example BN
P(A) = 0.001
a
P(B|A) = 0.3
P(B|~A) = 0.001
b
P(C|A) = 0.2
P(C|~A) = 0.005
c
d
P(D|B,C) = 0.1
P(D|B,~C) = 0.01
P(D|~B,C) = 0.01
P(D|~B,~C) = 0.00001
e
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 nondescendants 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 and chaining
• Independence assumption
– P ( x i |  i , q)  P ( x i |  i )
i
where q is any set of variables
q
(nodes) other than x i and its successors
xi
–  i blocks influence of other nodes on x i
and its successors (q influences x i only
through variables in  i )
– 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
P ( x1 ,..., xn )   ni1 P ( xi |  i )
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Chaining: Example
a
b
c
d
e
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)
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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
a
b
•
•
•
•
c
d
e
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
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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
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–
–
–
–
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Stochastic simulation / sampling methods
Markov chain Monte Carlo methods
Genetic algorithms
Neural networks
Simulated annealing
Mean field theory
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