The BIPF algorithm is a Markovian algorithm with the purpose of simulating certain probability distributions supported by contingency tables belonging to hierarchical log-linear models. The updating steps of the algorithm depend only on the required expected marginal tables over the maximal terms of the hierarchical model. Usually these tables are marginals of a positive joint table, in which case it is well known that the algorithm is a blocking Gibbs Sampler. But the algorithm makes sense even when these marginals do not come from a joint table. In this case the target distribution of the algorithm is necessarily improper. In this paper we investigate the simplest non trivial case, i. e. the 2×2×2 hierarchical interaction. Our result is that the algorithm is asymptotically attracted by a limit cycle in law.
The paper deals with practical aspects of decision making under uncertainty on finite sets. The model is based on {\em marginal problem}. Numerical behaviour of 10 different algorithms is compared in form of a study case on the data from the field of rheumatology. (Five of the algorithms types were suggested by A. Perez.) The algorithms (expert systems, inference engines) are studied in different {\em situations} (combinations of parameters).
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.
A possibilistic marginal problem is introduced in a way analogous to probabilistic framework, to address the question of whether or not a common extension exists for a given set of marginal distributions. Similarities and differences between possibilistic and probabilistic marginal problems will be demonstrated, concerning necessary condition and sets of all solutions. The operators of composition will be recalled and we will show how to use them for finding a T-product extension. Finally, a necessary and sufficient condition for the existence of a solution will be presented.