Markov Networks: Theory and Applications Ying Wu Electrical Engineering and Computer Science Northwestern University Evanston, IL

2 Warm up … Given a directed graphical model Prove x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 p(x 2 |x 1 ) p(x 4 |x 2,x 3 ) p(x 6 |x 3 ) p(x 1 ) p(x 3 ) p(x 5 |x 4,x 6 )

3 Wake up ! Given a HMM: p(Z i |X i ) and p(X t |X t-1 ) Prove: x1x1 x2x2 xkxk x t-1 xtxt z1z1 z2z2 zkzk z t-1 ztzt …

4 Interesting problems (i) ? Super-resolution –Given a low-res image, can you recover a high-res image? –To make it feasible, you may be given a set of training pairs of low-res and high-res images.

5 Interesting problems (ii) Capturing body motion from video –Given a video sequence, you need to estimate the articulated body motion from the video.

6 Outline Markov network representation Three issues –Inferring hidden variables –Calculating evidence –Learning Markov networks Inference –Exact inference –Approximate inference Applications –Super-resolution –Capturing body articulation

7 Representations Undirected graphical models –G(V, E), v  V represents a random variable x, e  E indicate the relation of two random variables; –Each r.v. x i has a prior  i (x i ); –Each undirected link (between x i and x j ) is associated with a potential function  ij (x i, x j ); –v  V, N(v) means the neighborhood of v; We can combine undirected and directed graphical models.

8 Markov Network Undirected links Directed links

9 Three issues Inferring hidden variables –Given a Markov network  –To infer the hidden variable of the evidence p(X|Z,  ) Calculating evidence –Given a Markov network  –To calculate the likelihood of the evidence p(Z|  ) Learning Markov networks –Given a set of training data {Z 1,…,Z n } –To find the best model

10 To be covered The inference problem is the key among the three –Why? –Let’s discuss … We will focus on inference in this lecture

11 An example Given the above model, with –p(z i |x i ),  i (x i ) –  12 (x 1,x 2 ),  23 (x 2,x 3 ),  34 (x 3,x 4 ),  45 (x 4,x 5 ),  46 (x 4,x 6 ) To solve –p(x 3 |z 1,z 2,z 3,z 4,z 5,z 6 ) x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 z1z1 z2z2 z3z3 z4z4 z5z5 z6z6

12 An example “message” x 2  x 3 “message” x 4  x 3

13 What do we learn from the Ex.? x1x1 x2x2 x3x3 x4x4 x5x5 x6x6 z1z1 z2z2 z3z3 z4z4 z5z5 z6z6 M 12 (x 2 ) M 23 (x 3 ) M 43 (x 3 ) M 54 (x 4 ) M 64 (x 4 )

14 What do we learn from the Ex.?

15 Sum-Product Algorithm Message passing xkxk xixi M ki (x i ) zkzk  ik (x i, x k )

16 Properties Properties of “belief propagation” –When the network is not loopy, the algorithm is exact; –When the network is loopy, the inference can be done by iterating the message passing procedures; –In this case, the inference is not exact, –and it is not clear how good is the approximation.

17 Application: Super-resolution Bill Freeman (ICCV’99) –X: high-frequency data –Z: mid-frequency data –“scene”: the high-frequency components –“evidence”: the mid-freq of the low-res image input –The “scene” is decomposed by a network of scene patches; and each scene patch is associated with an evidence patch –Learning p(z|x) from training samples scene patch evidence patch

18 Results (Freeman) + = Low-res input Iteration of B.P. to obtain high-frequency data Recovered high-res image actual high-res actual high-freq

19 Results (Freeman) Training data

20 Variational Inference We want to infer It is difficult, because of the networked structure. We perform probabilistic variational analysis The idea is to find an optimal approximation of, such that the Kullback-Leibler (KL) divergence of these two distribution is minimized:

21 Mean Field Theory When we choose a full factorization variation: We end up with a very interesting result: a set of fixed point equations: This is very similar to the Mean Field theory in statistical physics.

22 Computational Paradigm Three factors affect the posterior of a node: –Local prior –Neighborhood prior –Local likelihood

23 Application: body tracking Capturing body articulation (Wu et al.) –X: the motion of body parts –Z: image features (e.g., edges) –The motion of body parts are constrained and correlated –Specifying  (x i,x j ) –modeling p(z|x) explicitly

24 Dynamic Markov Network

25 Mean Field Iterations We are still investigating the convergence property of mean field algorithms But our empirical study shows that it converges very fast, generally, less than five iterations.

26 Application: body tracking

27 Application: body tracking

28 Summary Markov networks contain undirected edges in the graph to model the non-casual correlation Inference is the key of analyzing Markov networks –Exact inference –Approximate inference Two inference techniques were introduced here: –Belief propagation –Variational inference Two applications –Super-resolution –Capturing articulated body motion