Bayesian analysis with limited communication
The i-th member of a group of m individuals (or stations) observes a random quantity Xi, where X=(X1,...,Xm) has a density g(x |π). Each individual can report only yi=hi(xi), because of a limitation on the amount of information that can be communicated. Based on y = (y1,...,ym) and a prior distribution π(θ), Bayesian inference or decision concerning θ 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 π and the reports, yi, are the posterior distributions of each team member. We compare the optimal Bayesian posterior for this problem (π(θ | 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 yi can be only 0 or 1, and will illustrate the optimal choice of yi for both inference and decision criteria. The inference criterion considered will be closeness of the posteriors π(θ | x) and π(θ | 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. © 1991.
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