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 {PA;A∈\SS}, where \SS is a class of subsets of N with ⋃\SS=N. The paper deals with the problem of approximating P on the basis of those given marginals. The divergence of an approximation P^ from P is measured by the relative entropy H(P|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 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 P^ with prescribed marginals is recalled. If the condition holds then both methods for approximating P give the same result.
An overview is given of results achieved by F. Matúš on probabilistic conditional independence (CI). First, his axiomatic characterizations of stochastic functional dependence and unconditional independence are recalled. Then his elegant proof of discrete probabilistic representability of a matroid based on its linear representability over a finite field is recalled. It is explained that this result was a basis of his methodology for constructing a probabilistic representation of a given abstract CI structure. His embedding of matroids into (augmented) abstract CI structures is recalled and his contribution to the theory of semigraphoids is mentioned as well. Finally, his results on the characterization of probabilistic CI structures induced by four discrete random variables and by four regular Gaussian random variables are recalled. Partial probabilistic representability by binary random variables is also mentioned., Milan Studený., and Obsahuje bibliografické odkazy
In this paper we study two operations of merging components in a chain graph, which appear to be elementary operations yielding an equivalent graph in the respective sense. At first, we recall basic results on the operation of {\em feasible merging} components, which is related to classic LWF (Lauritzen, Wermuth and Frydenberg) Markov equivalence of chain graphs. These results are used to get a graphical characterisation of {\em factorisation equivalence} of classic chain graphs. As another example of the use of this operation, we derive some important invariants of LWF Markov equivalence of chain graphs. Last, we recall analogous basic results on the operation of {\em legal merging} components. This operation is related to the so-called {\em strong equivalence} of chain graphs, which includes both classic LWF equivalence and alternative AMP (Andersson, Madigan and Perlman) Markov equivalence.