In This Exercise We Will Derive A Message Passing Algorithm Called The Child Parent 2528120

In this exercise, we will derive a message passing algorithm, called the child-parent algorithm, for Bethe region graphs. Although the messages in this algorithm are somewhat convoluted, they have the advantage of corresponding directly to the Lagrange multipliers for the region graph energy functional. Moreover, although somewhat opaque, the message passing algorithm is a very slight variation on the scheme used in standard belief propagation: the standard messages are simply raised to a power. Consider the Bethe cluster, where we assume that all counting numbers are nonzero. Let ρi = 1/κi and ρr = 1/κr. Starting from equation (11.34), derive the correctness of the following update rules for the messages and the beliefs. For all r ∈ R+: