Message Passing in Cluster Graphs

Message Passing in a Cluster Graph. 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 i,j (Xi, Xj), with one potential for each pair of variables Xi, Xj connected by an edge in M. Which of the following expressions correctly computes the message 6345 that cluster C3 will send to cluster C6 during belief propagation? Assume that the variables in the sepsets are equal to the intersection of the variables in the adjacent cliques. O 5345(X5) = 2X2 2,5(X2,X5) O 53+5(X5) 2 2X2 452,5(X2,X5)52>3(X2l54>3(X2)57>3(X5)56>3(X2) O 53)5(X2) 2 2X5 452,5(X2,X5l52>3(X2l54>3(X2)57>3(X5) .63>6(X5) 2 2X2 452,5(X2,X5)52>3(X2l54>3(X2)57>3(X5) Message Passing Computation. Consider the Markov network M from the previous question. If the initial factors in the 1 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 63_,6, 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 1. Separate the entries of the message with spaces. Order the entries by lexicographic variable order: for example, if the message is over one variable X,, then enter in 63_,6 (X, = O) 63_,5(X,> = 1). If the message is over two variables Xi, Xj ,wherei