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Belief Propagation. Kai Ju Liu March 9, 2006. Statistical Problems. Medicine Finance Internet Computer vision. Inference Problems. Given data B , infer A : p ( A | B ) Computer vision Given image, find objects Given two images, resolve 3D object Given multiple images, track object.
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Belief Propagation Kai Ju Liu March 9, 2006
Statistical Problems • Medicine • Finance • Internet • Computer vision
Inference Problems • Given data B, infer A: p(A|B) • Computer vision • Given image, find objects • Given two images, resolve 3D object • Given multiple images, track object
A B Conditional Probability • Given event B, what is probability of A? • Independence: p(A|B)=p(A)
Cold Weekday Party Hangover Joint Probability: Marginal Probability: 8-sum e.g.
Cold Weekday Party Hangover Marginal Probability: 8-sum Localize probabilities: e.g. (cont.)
Approach • Define variables and connections • Calculate marginal probabilities efficiently • Find most likely configuration
Pairwise Markov Random Field 4 1 2 3 5 • Basic structure: vertices, edges
and observed value yi • Compatibility between states and observed values, • Compatibility between neighboring vertices i and j, Pairwise Markov Random Field • Basic structure: vertices, edges • Vertex i has set of possible states Xi
Marginal probability: Pairwise MRF: Probabilities • Joint probability: • Advantage: allows average over ambiguous states • Disadvantage: complexity exponential in number of vertices
Belief Propagation 4 1 2 3 5
Messages propagate information: Belief Propagation • Beliefs replace probabilities:
Belief Propagation Example 4 1 3 5
When can we calculate beliefs exactly? When do beliefs equal probabilities? When is belief propagation efficient? Answer: Singly-Connected Graphs (SCG’s) • Graphs without loops • Messages terminate at leaf vertices • Beliefs equal probabilities • Complexity in previous example reduced from 13S5 to 24S2 BP: Questions
Messages do not terminate Possible approximate solutions Standard belief propagation Generalized belief propagation BP-TwoGraphs: Goals • Utilize advantages of SCG’s • Be accurate and efficient on loopy graphs BP on Loopy Graphs
Calculate beliefs on each set of SCG’s: • Select maximum beliefs from both sets BP-TwoGraphs: SCG’s • Consider loopy graph with n vertices • Select two sets of SCG’s that approximate the graph
BP-TwoGraphs: Vision SCG’s • Rectangular grid of pixel vertices • Hi: horizontal graphs • Gi: vertical graphs
Image Segmentation add noise segment
