Partially ordered models

TL;DR

This paper introduces partially ordered chains (POC) as a new mathematical framework that generalizes probabilistic cellular automata and models textures and independence in statistics, with foundational properties and uniqueness criteria.

Contribution

It provides a formal definition of POC, explores their properties, and establishes three criteria for their uniqueness, bridging concepts from cellular automata, Gibbs measures, and statistical mechanics.

Findings

Defined POC using partially ordered specifications

Established basic geometrical and probabilistic properties

Proved three criteria for uniqueness of POC

Abstract

We provide a formal definition and study the basic properties of partially ordered chains (POC). These systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics (in the later case they are known as Bayesian networks). Our chains are a generalization of probabilistic cellular automata (PCA) and their theory has features intermediate between that of discrete-time processes and the theory of statistical mechanical lattice fields. Its proper definition is based on the notion of partially ordered specification (POS), in close analogy to the theory of Gibbs measure. This paper contains two types of results. First, we present the basic elements of the general theory of POCs: basic geometrical issues, definition in terms of conditional probability kernels, extremal decomposition, extremality and triviality, reconstruction starting from single-site kernels, relations between POM and Gibbs fields. Second, we prove three uniqueness criteria that correspond to the criteria known as bounded uniformity, Dobrushin and disagreement percolation in the theory of Gibbs measures.