Introduction to Belief Propagation

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

4. HMM(Hidden Markov Model) • Find probabilities of states with given observations.

5. MRF(Markov Random Field) • HMM • MRF

6. MRF(Markov Random Field)

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

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

9. 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)\jmki(xi) • Messages initially uniformly distributed MLRG

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

11. 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) MLRG

12. Optimization for MRF • Convert to energy domain • Maximizing

13. Definitions of Message and Belief mij(xj) = (xi) (xi, yi)(xi, xj) kN(i)\jmk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 • http://www.ski.org/Rehab/Coughlan_lab/General/TutorialsandReference/BPtutorial.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/seminars/waldmarkov.pdf