B.S student YeongWon Kim

Introduction to Belief Propaga-tion

Markov Property

Markov Chain Hidden Markov Model Markov Random Field Belief Propagation

Markov Process

Markov Chain

Day 1 2 3 4 5

Rainy 1 ? ? ? ?

Sunny 0 ? ? ? ?

Find probabilities of states with given ob-servations.

HMM(Hidden Markov Model)

Day 1 2 3 4 5

Observation Walk Walk Shop Clean Shop

Rainy ? ? ? ? ?

Sunny ? ? ? ? ?

𝑥𝑛−1 𝑥𝑛 𝑥𝑛+1

HMM

MRF

MRF(Markov Random Field)

𝑥𝑛−1 𝑥𝑛 𝑥𝑛+1

Day 1 2 3 4 5

Observation Walk Walk Shop Clean Shop

Rainy ? ? ? ? ?

Sunny ? ? ? ? ?

MRF(Markov Random Field)

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Question: What are the marginal distribu-tions for xi, i = 1, …,n?

MRF formulation

3/1/2008 MLRG

x1 x2

xi

xn

y1 y2

yi

yn

P(x1, x2, …, xn) = (1/Z) (ij) (xi, xj) i (xi, yi)

Belief– Marginal distribution

Message– Joint distribution

-Sum-product-Max-product

Belief Propagation

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Message mij from xi to xj : what node xi thinks about the marginal distribu-tion of xj

Message Updating

3/1/2008 MLRG

xi xj

yi yj

N(i)\j

mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)

Messages initially uniformly distributed

Message Updating

L1 L1

L2

L3

Ln

L2

L3

Node P Node Q

Ln

mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)

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Belief

3/1/2008 MLRG

xj

yj

N(j)

b(xj) = k (xj, yj) qN(j) mqj(xj)

Belief b(xj): what node xj thinks its marginal distribu-tion is

Convert to energy domain

Maximizing

Optimization for MRF

Definitions of Message and Belief

P(x1, x2, …, xn) = (1/Z) (ij) (xi, xj) i (xi, yi)

mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)

b(xj) = k (xj, yj) qN(j) mqj(xj)

1. Initialize all messages uniformly.2. For i from 1 to number of iterations3. Update all messages.4. End5. For each nodes, find a label that has max-

imum belief.

Pseudocode

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Result

http://www.ski.org/Rehab/Coughlan_lab/General/TutorialsandReference/BPtutori-al.pdf

http://homepages.inf.ed.ac.uk/rbf/CVonline/LOCAL_COPIES/AV0809/ORCHARD/

Wikipedia Efficient Belief Propagation for Early Vision Understanding Belief Propagation and its

Generalizations http://www.stats.ox.ac.uk/~steffen/semi-

nars/waldmarkov.pdf

Reference