Download 3.1.1-Inf-Overview-Prob

Probabilistic
Graphical
Models
Inference
Overview
Conditional
Probability
Queries
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Inference in a PGM
• PGM encodes P(X1,…,Xn)
• Can be used to answer any query over
joint distribution
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Conditional Probability Queries
• Evidence: E = e
• Query: a subset of variables Y
• Task: compute P(Y | E=e)
• Applications
– Medical/fault diagnosis
– Pedigree analysis
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NP-Hardness
The following are all NP-hard
• Given a PGM P, a variable X and a value
xVal(X), compute P(X=x)
– Or even decide if P(X=x) > 0
• Let  < 0.5. Given a PGM P, a variable X and a
value xVal(X), and observation eVal(E), find a
number p that has |P(X=x|E=e) – p| < 
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Sum-Product
C
I
D
G
S
L
H
J
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Sum-Product
A
D
B
C
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Evidence: Reduced Factors
A
D
B
C
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Evidence: Reduced Factors
C
P(J,I=i, H=h) =
I
D
G
S
L
H
J
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Sum-Product
Compute
and renormalize
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Algorithms: Conditional Probability
• Push summations into factor product
– Variable elimination
• Message passing over a graph
– Belief propagation
– Variational approximations
• Random sampling instantiations
– Markov chain Monte Carlo (MCMC)
– Importance sampling
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Summary
• Conditional probability queries of subset of
variables given evidence on others
• Summing over factor product
• Evidence simply reduces the factors
• Many exact and approximate algorithms
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