Composition of Probability Measures on Finite Spaces

TL;DR

This paper establishes foundational concepts for probabilistic models on finite spaces, focusing on composition operators, perfect sequences, and the potential for unifying various graph models like Bayesian networks.

Contribution

It introduces the concept of perfect sequences and lays the groundwork for a unified approach to decomposable models and Bayesian networks.

Findings

Introduction of perfect sequences for probabilistic models

Framework for changing operator orderings in composition sequences

Potential unification of graph models through composition operators

Abstract

Decomposable models and Bayesian networks can be defined as sequences of oligo-dimensional probability measures connected with operators of composition. The preliminary results suggest that the probabilistic models allowing for effective computational procedures are represented by sequences possessing a special property; we shall call them perfect sequences. The paper lays down the elementary foundation necessary for further study of iterative application of operators of composition. We believe to develop a technique describing several graph models in a unifying way. We are convinced that practically all theoretical results and procedures connected with decomposable models and Bayesian networks can be translated into the terminology introduced in this paper. For example, complexity of computational procedures in these models is closely dependent on possibility to change the ordering of oligo-dimensional measures defining the model. Therefore, in this paper, lot of attention is paid to possibility to change ordering of the operators of composition.