Reversible Imprecise Markov Chains

• Introduces a general framework for time reversal and reversibility in imprecise Markov chains under the strong independence interpretation. • Proposes a symmetric representation of imprecise Markov chains based on sets of two-step joint distribution matrices. • Demonstrates that reversibility of an imprecise Markov chain corresponds to symmetry of its joint distribution set. • Establishes the relation with random walks on weighted graphs, where edge weights are given by intervals and no symmetry of individual walks is required. • Formulates the evaluation of expectation bounds for n-step functionals as a nonlinear optimization problem. 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 complete 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 discuss computational approaches for evaluating lower and upper expectations of such functionals.