Foundations of compositional models: structural properties

Abstract

The paper is a follow-up of [R.J.: Foundations of compositional model theory. IJGS, 40(2011): 623-678], where basic properties of compositional models, as one of the approaches to multidimensional probability distributions representation and processing, were introduced. In fact, it is an algebraic alternative to graphical models, which does not use graphs to represent conditional independence statements. Here, these statements are encoded in a sequence of distributions to which an operator of composition -the key element of this theory -is applied in order to assemble a multidimensional model from its low-dimensional parts. In this paper, we show a way to read conditional independence relations, and to solve related topics, above all the socalled equivalence problem, i.e. the problem of recognizing whether two different structures induce the same system of conditional independence relations.

Compositional models, Bayesian models and recursive factorization models

“…DOI: 10.14736/kyb-2016-5-0696 Our result is stronger than the result proven by Jirousek and Kratochvíl (see Section 5 in [9]) which states that, for every Bayesian model, there exists a sequential expression that generates the same probability distributions represented by the Bayesian model. In order to prove this sort of equivalence between compositional models generated by sequential expressions and Bayesian models, we introduce recursive factorization models and prove that they are equivalent to both compositional models generated by sequential expressions and Bayesian models.…”

“…Compositional models of sequential type [2,4,5,6,8,9] were originally introduced to construct probability distributions from lower-order probability distributions as an operational alternative to Bayesian models [3] (also called "directed Markov models" [16]). Compositional models were also applied to belief functions [7,11,12], possibility functions [11] and Shenoy valuations [10].…”

Marginalization in models generated by compositional expressions

“…We leave to future research the comparison of algebraic equivalence with the notion of equivalence introduced in [27] and with "Markov equivalence" [13,22].…”

“…Compositional models were introduced to construct probability distributions from lowerorder probability distributions [6,8,9,11,13] as an operational alternative to the graphical approach used to model Bayesian and causal networks. They were also applied to belief functions [10,15,16], possibility functions [15] and Shenoy valuations [14].…”