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

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

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.

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

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

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):

Example 4/ G

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

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