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Tree and local computations in a cross-entropy minimization problem with marginal constraints. (English) Zbl 1204.93113
Summary: In probability theory, Bayesian statistics, artificial intelligence and database theory the minimum cross-entropy principle is often used to estimate a distribution with a given set \(P\) of marginal distributions under the proportionality assumption with respect to a given “prior” distribution \(q\). Such an estimation problem admits a solution if and only if there exists an extension of \(P\) that is dominated by \(q\). In this paper we consider the case that \(q\) is not given explicitly, but is specified as the maximum-entropy extension of an auxiliary set \(Q\) of distributions. There are three problems that naturally arise: (1) the existence of an extension of a distribution set (such as \(P\) and \(Q\)), (2) the existence of an extension of \(P\) that is dominated by the maximum entropy extension of \(Q\), (3) the numeric computation of the minimum cross-entropy extension of \(P\) with respect to the maximum entropy extension of \(Q\). In the spirit of a divide-and-conquer approach, we prove that, for each of the three above-mentioned problems, the global solution can be easily obtained by combining the solutions to subproblems defined at node level of a suitable tree.

MSC:
93E10 Estimation and detection in stochastic control theory
62E10 Characterization and structure theory of statistical distributions
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