Kybernetika 43 no. 5, 591-618, 2007

Comparison of two methods for approximation of probability distributions with prescribed marginals

Albert Perez and Milan Studený


Let $P$ be a discrete multidimensional probability distribution over a finite set of variables $N$ which is only partially specified by the requirement that it has prescribed given marginals $\{ P_{A}; A\in \SS\}$, where $\SS$ is a class of subsets of $N$ with $\bigcup\SS = N$. The paper deals with the problem of approximating $P$ on the basis of those given marginals. The divergence of an approximation $\hat{P}$ from $P$ is measured by the relative entropy $H(P|\hat{P})$. Two methods for approximating $P$ are compared. One of them uses formerly introduced concept of {\em dependence structure simplification\/} (see Perez \cite{Per79}). The other one is based on an {\em explicit expression}, which has to be normalized. We give examples showing that neither of these two methods is universally better than the other. If one of the considered approximations $\hat{P}$ really has the prescribed marginals then it appears to be the distribution $P$ with minimal possible multiinformation. A simple condition on the class $\SS$ implying the existence of an approximation $\hat{P}$ with prescribed marginals is recalled. If the condition holds then both methods for approximating $P$ give the same result.