1. Introduction to Belief Propagation
B.S student YeongWon Kim

2. Markov Process Markov Property Markov Chain Hidden Markov Model
Markov Random Field
Belief Propagation

3. Markov Chain
Day 1 2 3 4 5
Rainy ?
Sunny

4. HMM(Hidden Markov Model)
Find probabilities of states with given observations.
Day 1 2 3 4 5
Observation
Walk
Shop
Clean
Rainy ?
Sunny

5. MRF(Markov Random Field)
HMM
MRF
Day 1 2 3 4 5
Observation
Walk
Shop
Clean
Rainy ?
Sunny
𝑥 𝑛−1
𝑥 𝑛
𝑥 𝑛+1
𝑥 𝑛−1
𝑥 𝑛
𝑥 𝑛+1

6. MRF(Markov Random Field)

7. P(x1, x2, …, xn) = (1/Z) (ij) (xi, xj) i (xi, yi)
MRF formulation
Question: What are the marginal distributions for xi, i = 1, …,n? x1 x2 xi xn y1 y2 yi yn
P(x1, x2, …, xn) = (1/Z) (ij) (xi, xj) i (xi, yi)
3/1/2008
MLRG

8. Belief Propagation Belief Message -Sum-product -Max-product
Marginal distribution
Message
Joint distribution
-Sum-product
-Max-product

9. mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)
Message Updating
Message mij from xi to xj : what node xi thinks about the marginal distribution of xj yi yj
N(i)\j xi xj
mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)
Messages initially uniformly distributed
3/1/2008
MLRG

10. mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)
Message Updating
Node P
Node Q L1 L1 L2 L2 L3 L3
mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi) Ln Ln

11. b(xj) = k (xj, yj) qN(j) mqj(xj)
Belief
Belief b(xj): what node xj thinks its marginal distribution is
N(j) yj xj
b(xj) = k (xj, yj) qN(j) mqj(xj)
3/1/2008
MLRG

12. Optimization for MRF Convert to energy domain
Maximizing 𝐹 𝑥1,𝑥2 +𝐺 𝑥2 +𝐻 𝑥2,𝑥3,𝑥4 .

13. Definitions of Message and Belief
mij(xj) = (xi) (xi, yi) (xi, xj) kN(i)\j mki(xi)
b(xj) = k (xj, yj) qN(j) mqj(xj)
P(x1, x2, …, xn) = (1/Z) (ij) (xi, xj) i (xi, yi)

14. Pseudocode Initialize all messages uniformly.
For i from 1 to number of iterations
Update all messages.
End
For each nodes, find a label that has maximum belief.

15. Result

16. Reference
Wikipedia
Efficient Belief Propagation for Early Vision
Understanding Belief Propagation and its Generalizations