Reversible Imprecise Markov Chains

TL;DR

This paper develops a theoretical framework for reversible imprecise Markov chains, emphasizing their structural properties, symmetric representation, and implications for path-dependent expectations, extending classical reversible Markov chain theory.

Contribution

It introduces a symmetric credal set representation for reversible imprecise Markov chains, enabling reversal operations and unified analysis of forward and reverse dynamics.

Findings

Reversible imprecise Markov chains can be represented by symmetric credal sets.

Reversibility reduces to matrix symmetry in the proposed framework.

The framework allows computation of bounds for path-dependent expectations.

Abstract

Reversible Markov chains play a central role in stochastic modelling and in algorithms such as Markov chain Monte Carlo (MCMC). Motivated by the fundamental importance of reversibility in classical settings, this paper develops a theoretical framework for reversible imprecise Markov chains. We focus on their structural properties and their representation through joint distribution matrices. Adopting the strong independence interpretation, we reverse every precise chain compatible with a given imprecise Markov chain specification. Since the reversed ensemble generally cannot be encoded by the usual forward model defined by an imprecise initial distribution and a set of transition matrices, we introduce a symmetric representation based on credal sets of two-step joint distribution (or edge measure) matrices. This strictly more expressive framework naturally admits the reversal operation and reduces reversibility to simple matrix symmetry. Moreover, forward and reverse dynamics can be described simultaneously within a single closed convex set, providing a unified structural basis for the analysis of expectations of path-dependent functionals. We illustrate the theory with random walks on graphs and outline methods for computing lower and upper expectations of such functionals.