As said by Mareš and Mesiar, necessity of aggregation of complex real inputs appears almost in any field dealing with observed (measured) real quantities (see the citation below). For aggregation of probability distributions Sklar designed his copulas as early as in 1959. But surprisingly, since that time only a very few literature have appeared dealing with possibility to aggregate several different pairwise dependencies into one multivariate copula. In the present paper this problem is tackled using the well known Iterative Proportional Fitting Procedure. The proposed solution is not an exact mathematical solution of a marginal problem but just its approximation applicable in many practical situations like Monte Carlo sampling. This is why the authors deal not only with the consistent case, when the iterative procedure converges, but also with the inconsistent non-converging case. In the latter situation, the IPF procedure tends to cycle (when combining three pairwise dependencies the procedure creates three convergent subsequences), and thus the authors propose some heuristics yielding a "solution'' of the problem even for inconsistent pairwise dependence relations.
Perez's approximations of probability distributions by dependence structure simplification were introduced in 1970s, much earlier than graphical Markov models. In this paper we will recall these Perez's models, formalize the notion of a compatible system of elementary simplifications and show the necessary and sufficient conditions a system must fulfill to be compatible. For this we will utilize the apparatus of compositional models.
Efficient computational algorithms are what made graphical Markov models so popular and successful. Similar algorithms can also be developed for computation with compositional models, which form an alternative to graphical Markov models. In this paper we present a theoretical basis as well as a scheme of an algorithm enabling computation of marginals for multidimensional distributions represented in the form of compositional models.
The knowledge of causal relations provides a possibility to perform predictions and helps to decide about the most reasonable actions aiming at the desired objectives. Although the causal reasoning appears to be natural for the human thinking, most of the traditional statistical methods fail to address this issue. One of the well-known methodologies correctly representing the relations of cause and effect is Pearl's causality approach. The paper brings an alternative, purely algebraic methodology of causal compositional models. It presents the properties of operator of composition, on which a general methodology is based that makes it possible to evaluate the causal effects of some external action. The proposed methodology is applied to four illustrative examples. They illustrate that the effect of intervention can in some cases be evaluated even when the model contains latent (unobservable) variables.
Several systems supporting development and application of graphical
Markov inodels are widely used; perliaps the most famous are HUGIN and NETICA, which are supporting Bayesian networks. The goal of this paper is to introduce system MUDIM, which is intended to support non-graphical multidimensional models, namely cornpositional models. The basic idea of these inodels resembles jig-saw puzzle, where a picture must be assembled from a great number of pieces, each bearing a small part of a picture. Analogously, compositional models of a multidimensional distribution are assenililed (composed) of a great number of low-dirnensional distributions.
One of the advantages of this approach is that the same apparatus that is based on operators of composition, can be applied for description of both probabilistic and possibilistic models. This is also the goal for future MUDIM development, to extend it in the way that it will be able to process both probabilistic and possibilistic models.
In contrast to most other approaches used to represent multidimensional probability distributions, which are based on graphical Markov modelling (i.e. dependence structure of distributions is represented by graphs), the described method is rather procedural. Here, we describe a process by which a multidimensional distribution can be composed from a “generating sequence” - a sequence of low-dimensional distributions. The main advantage of this approach is that the same apparatus based on operators of composition can be applied for description of both probabilistic and possibilistic models.