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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:
 9.3e+11 Estimation and detection in stochastic control theory 6.2e+11 Characterization and structure theory of statistical distributions
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