Analyze which messages change after we update the

Question

Problem

Suppose that we have an incomplete data set D, and network structure G and matching parameters. Moreover, suppose that we are interested in learning the parameters of a single CPD P(X_i| U_i). That is, we assume that the parameters we were given for all other families are frozen and do not change during the learning. This scenario can arise for several reasons: we might have good prior knowledge about these parameters; or we might be using an incremental approach, as mentioned in box 19.C. We now consider how this scenario can change the computational cost of the EM algorithm.

a. Assume we have a clique tree for the network G and that the CPD P(X_i|U_i) was assigned to clique C_j. Analyze which messages change after we update the parameters for P(X_i| U_i). Use this analysis to show how, after an initial pre-computation step, we can perform iterations of this single-family EM procedure with a computational cost that depends only on the size of Cj and not the size of the rest of the cluster tree.

b. Would this conclusion change if we update the parameters of several families that are all assigned to the same cluster in the cluster tree?