BAYESIAN-ANALYSIS WITH LIMITED COMMUNICATION

The i-th member of a group of m individuals (or stations) observes a random quantity X(i), where X = (X(l),...,X(m) has density g(x\theta). Each individual can report only y(i) = h(i)(x(i)), because of a limitation on the amount of information that can be communicated. Based on y = (y(l),...,y(m)) and a prior distribution pi(theta), Bayesian inference or decision concerning theta is to be undertaken. The first version of this problem that will be studied is the 'team' problem, where the m individuals form a team with common prior pi and the reports, y(i), are the posterior distributions of each team member. We compare the optimal Bayesian posterior for this problem (pi(theta\y)) with previous suggestions, such as the optimal linear opinion pool. The second facet of the problem that is explored is that of choosing y to optimize the information communicated, subject to a constraint on the amount of information that can be communicated. In particular, we will consider the dichotomous case, in which each y(i) can be only 0 or 1, and will illustrate the optimal choice of y(i) for both inference and decision criteria. The inference criterion considered will be closeness of the posteriors pi(theta\x) and pi(theta\y), in an expected Kullback-Leibler sense, while the decision criterion considered will be usual optimality with respect to overall expected loss. Examples are presented, including discussion of a situation that arises in reliability demonstration.