Foundations of compositional model theory

Abstract

Graphical Markov models, most of all Bayesian networks, have become a very popular way for multidimensional probability distribution representation and processing. What makes representation of a very-high-dimensional probability distribution possible is its independence structure, i.e. a system of conditional independence relations valid for the distribution in question. The fact that some of independence systems can be successfully represented with the help of graphs is reflected in the general title: graphical modelling. However, graphical representation of independence structures is also associated with some disadvantages: only a small part of different independence structures can be faithfully represented by graphs; and still one structure is usually equally well represented by several graphs. These reasons, among others, initiated development of an alternative approach, called here theory of compositional models, which enables us to represent exactly the same class of distributions as Bayesian networks. This paper is a survey of the most important basic concepts and results concerning compositional models necessary for reading advanced papers on computational procedures and other aspects connected with this (relatively new) approach for multidimensional distribution representation.