Daphne Koller

Overview

Maximum aposteriori (MAP)

ProbabilisticGraphicalModels

Inference

Daphne Koller

Maximum a Posteriori (MAP)• Evidence: E=e • Query: all other variables Y (Y={X1,…,Xn}-E)• Task: compute MAP(Y|E=e) = argmaxy P(Y=y | E=e)

– Note: there may be more than one possible solution

• Applications– Message decoding: most likely transmitted message– Image segmentation: most likely segmentation

Daphne Koller

MAP ≠ Max over Marginals• Joint distribution– P(a0, b0) = 0.04– P(a0, b1) = 0.36– P(a1, b0) = 0.3– P(a1, b1) = 0.3

BA B0 B1

a0 0.1 0.9a1 0.5 0.5

Ia0 a1

0.4 0.6

P(B|A)P(A)

A

B

Daphne Koller

NP-HardnessThe following are NP-hard• Given a PGM P, find the joint assignment x with highest probability P(x)

• Given a PGM P and a probability p, decide if there is an assignment x such that P(x) > p

Daphne Koller

Max-ProductC

D I

SG

L

JH

D

Daphne Koller

Evidence: Reduced Factors

BD

C

A

Daphne Koller

Max-Product

Daphne Koller

Algorithms: MAP• Push maximization into factor product– Variable elimination

• Message passing over a graph–Max-product belief propagation

• Using methods for integer programming• For some networks: graph-cut methods• Combinatorial search

Daphne Koller

Summary• MAP: single coherent assignment of

highest probability– Not the same as maximizing individual

marginal probabilities• Maxing over factor product• Combinatorial optimization problem• Many exact and approximate algorithms

Daphne Koller

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