Bayesian Belief Propagation for Image Understanding

1. Bayesian Belief Propagation for Image Understanding • David Rosenberg

2. Markov Random Fields • Let G be an undirected graph • nodes: {1, …, n} • Associate a random variable X_t to each node t in G. • (X_1, …, X_n) is a Markov random field on G if • Every r.v. is independent of its nonneighbors conditioned on its neighbors. • P(X_t=x_t | X_s = x_s for all s \neq t} = P(X_t=x_t | X_s = x_s for all s\in N(t)),where N(s) be the set of neighbors of a node s.

3. Specifying a Markov Random Field • Nice if we could just specify P( X | N(X) )for all r.v.’s X (as with Bayesian networks) • Unfortunately, this will overspecify the joint PDF. • E.g. X_1 -- X_2. • Joint PDF has 3 degrees of freedom • Conditiona PDFs X_1|X_2 and X_2|X_1 have 2 degrees of freedom each • The Hammersley-Clifford Theorem helps to specify MRFs

4. The Gibbs Distribution • A Gibbs distribution w.r.t. graph G is a probability mass function that can be expressed in the form • P(x_1, … , x_n) = Prod _ Cliques C V_C(x_1, .., x_n) • where V_C(x_1, …, x_n) depends only on those x_I in C. • We can combine potential functions into products from maximal cliques, so • P(x_1, … , x_n) = Prod _ MaxCliques C V_C(x_1, .., x_n) • This may be better in certain circumstances because we don’t have to specify as many potential functions

5. Hammersley Clifford Theorem • Let the r.v’s {X_j} have a positive joint probability mass function. • Then the Hammersley Clifford Theorem says that {X_j} is a Markov random field on graph G iff it has a Gibbs distirubtion w.r.t G. • Side Note: Hammserley and Clifford discovered this theorem in 1971, but they didn’t publish it because they kept thinking they should be able to remove or relax the positivity assumption. They couldn’t. Clifford published the result in 1990. • Specifying the potential functions is equivalent to specifying the joint probability distribution of all variables. • Now it’s easy to specify a valid MRF • still not easy to determine the degrees of freedom in the distribution (normalization)

6. A Typical MRF Vision Problem • We have • hidden “scene” variables: X_j • observed “image” variables Y_j • Given X_j, Y_j is independent of everything else • Show Picture • The Problems • Given: Some instantiations of the Y_j’s • Find: • The aposteriori distribution over the X_j’s • Find the MAP estimate for each X_j • Find the least squares estimate of each X_j

7. Straightforward Exact Inference • Given the joint PDF • typically specified using potential functions • We can just marginalize out to • get the aposteriori distribution for each X_j • We can immediately extract the • MAP estimate -- just the mode of the aposteriori distriubtion • Least squares estimate -- just the expected value of the aposteriori distribution

8. Inference by Message Passing • The resulting aposteriori distributions are exact for graphs without loops (Pearl?) • Weiss and Freeman show that for arbitrary graph topologies, when belief propagation converges, it gives the correct least squares estimate (I.e. posterior mean) • More results?

9. y1 x1 y2 x2 y3 x3 Derivation of belief propagation

10. y1 y3 y2 x1 x3 x2 The posterior factorizes

11. y1 y3 y2 x1 x3 x2 Propagation rules

12. y1 y3 y2 x1 x3 x2 Propagation rules

13. Belief, and message updates j = i j i

14. Optimal solution in a chain or tree:Belief Propagation • “Do the right thing” Bayesian algorithm. • For Gaussian random variables over time: Kalman filter. • For hidden Markov models: forward/backward algorithm (and MAP variant is Viterbi).p

15. y1 y3 y2 x1 x3 x2 Y ( x , x ) 1 3 No factorization with loops!

16. First Toy Examples • Show messages passed and beliefs at each stage • show convergence in x steps.

17. Where does Evidence Fit In?

18. The Cost Functional Approach • We can state the solution to many problems in terms of minimizing a cost functional. • How can we put this our MRF framework?

19. Slide on Weiss’s Interior/exterior Example • show graphs of convergence speed

20. Slide on Weiss’s Motion Detection

21. My own computer example taking the cost functional approach

22. Discussion of complexity issues with message passing • How long are messages • How many messages do we have to pass per iteration • How many iterations until convergence • Problem quickly becomes intractible

23. Mention some apprxomiate inference approaches

24. Slides on message passing with jointly gaussian distributions???

25. EXTRA SLIDES

26. Incorporating Evidence nodes into MRFs • We would like to have nodes that don’t change their beliefs -- they are just observations. • Can we do this via the potential functions on the non-maximal clique containing just that node? • I tink this is what they do in the Yair Weiss implementation • What if we don’t want to specify a potential function? Make it identically one, since it’s in a product.

27. From cost functional to transition matrix

28. From cost functional to update rule

29. From update rule to transition matrix

30. The factoriation into pair wise potentials -- good for general Markov networks

31. Other Stuff • For shorthand, we will write x = (x_1, …, x_n).