1. CS774. Markov Random Field : Theory and Application Lecture 06 Kyomin Jung KAIST Sep 22 2009

2. Dynamic programming on the Junction Tree to compute marginals and MAP of G G Junction Tree of G Ex) consider Independent set problem

3. Hugin algorithm to build a junction tree Triangulate the graph to make it chordal Construct a junction tree from the triangulated graph A graph is chordal if each of its cycles of four or more nodes has a chord, which is an edge joining two nodes that are not adjacent in the cycle.

4. Exponential Family Parametrization of positive MRFs, i.e. P[x]>0 for all x. Let denote a collection of potential functions defined on the cliques of G. Let be a vector of weights on these potent ials functions. An MRF with weight is defined by Where the partition function is

5. Lemmas 1. 2. Hence the log partition function is convex as a function of.

6. Convex combinations Let denote the set of all spanning trees of G. Let be an exponential parameter vector that represents a tree T, i.e. only for the edges of T. Let be a probability distribution over T(G):

7. Example 4/3 1 1 11 00 0 0 G

8. Upper bound on the log partition ftn By Jensen’s inequality we obtain that For all and such that

9. Upper bound on the log partition ftn Note : number of spanning tree is large  ex Cayley’s formula says that # of spanning tree of a complete graph is Optimizing over with fixed It has a global minimum, and it could be solved exactly by nonlinear programming. To be continued