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Reversible Imprecise Markov Chains

Damjan Škulj

TL;DR

This work extends reversibility from classical precise Markov chains to imprecise settings by developing a symmetric joint-distribution (edge-measure) framework under strong independence. Reversibility is characterized by symmetry of the two-step joint distribution set, enabling a time-reversal operation that does not rely on invertible marginals or positive transitions. The authors show how to represent forward and reverse dynamics within a single convex joint-set, derive matrix-based conditions for reversibility, and apply the theory to random walks on graphs, including interval-weight graphs. Computationally, lower and upper expectations of path functionals reduce to multilinear programs over joint matrices, enabling robust inference under uncertainty. The framework offers a principled basis for robust, time-symmetric analysis of stochastic systems with partial parameter information, with potential extensions to continuous time and large-scale networks.

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.

Reversible Imprecise Markov Chains

TL;DR

This work extends reversibility from classical precise Markov chains to imprecise settings by developing a symmetric joint-distribution (edge-measure) framework under strong independence. Reversibility is characterized by symmetry of the two-step joint distribution set, enabling a time-reversal operation that does not rely on invertible marginals or positive transitions. The authors show how to represent forward and reverse dynamics within a single convex joint-set, derive matrix-based conditions for reversibility, and apply the theory to random walks on graphs, including interval-weight graphs. Computationally, lower and upper expectations of path functionals reduce to multilinear programs over joint matrices, enabling robust inference under uncertainty. The framework offers a principled basis for robust, time-symmetric analysis of stochastic systems with partial parameter information, with potential extensions to continuous time and large-scale networks.

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.

Paper Structure

This paper contains 36 sections, 25 theorems, 86 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Notation
  4. Markov chains
  5. Basic model
  6. Stationary time-homogeneous chains
  7. Reversible Markov chains
  8. Time reversal for time-inhomogeneous chains
  9. Joint-Distribution Matrices for Markov Processes
  10. Definition and basic identities
  11. Application to inhomogeneous chains
  12. Imprecise Markov chains
  13. Modelling imprecision in probability distributions
  14. Modelling imprecision in Markov chains
  15. Imprecise Markov chains as sets of processes
  16. ...and 21 more sections

Key Result

Theorem 1

Let $(X_n)_{n\in\mathbb{N}}$ be a stationary Markov chain on the finite state space $\mathcal{X}$ with transition matrix $P$ and stationary distribution $\pi$. The following statements are equivalent: Hence the chain is reversible if and only if any (and therefore all) of these conditions hold.

Theorems & Definitions (58)

  • Theorem 1
  • Definition 1: $q$-reverse
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Definition 2: Reverse transition law
  • Remark 1
  • Theorem 2
  • proof
  • ...and 48 more