6- Suppose we wish to perform inference over the Markov network M as shown below

Question

6- Suppose we wish to perform inference over the Markov network M as shown below Each of the variables Xi are binary, and the only potentials in the network are the pairwise potentials ?? (Xi-Xj), with one potential for each pair of variables Xi-X connected by an edge in M. (20 points) x,, x X2, X C2 C3 C, ) Write the expression to compute the message 36 that cluster C3 will send to cluster C6 during helie to the intersection of the variables in the adjacent cliques f propagation? Assume that the variables in the sepsets are equal b) If the initial factors in the Markov network M are of the form as shown in the table below, regardless of the specific value of i,j (we basically wish to encourage variables that are connected by an edge to share the same assignment), compute the message 036 (for all of the possible values of its scope), assuming that it is the first message passed during in loopy belief propagation. Assume that the messages are all initialized to the 1 message, i.e. all the entries are initially set to l 10 c Given that you can renormalize the messages at any point during belief propagation and still obtain correct marginals, consider the message Ö36 that you computed. Use this observation to compute the final and possibly approximate marginal probability Pix,-1 , X,-1) in cluster do at convergence (as extracted from the cluster beliefs)

Explanation / Answer

Markov Network

Markov Network (Markov Random Field) is an undirected graphical representation.
It is useful to model problems where the interaction between neighbours is undirected (W-X-Z-Y-W).

As in Bayesian Network (BN), nodes in Markov Network(MN) represent random variables, and edges represent interaction.

MN factorizes into Potential Functions.

Potentials in MN are associated with Cliques (C)-a complete subgraph/fully connected subgraph, and they are non-negative((C)>0).

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