Tree and local computations in a cross-entropy minimization problem with marginal constraints

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.