Imprecise Markov chains and their limit behaviour

TL;DR

This paper introduces imprecise Markov chains using credal sets to model uncertainty in probabilities, and demonstrates their long-term behavior converges to an invariant credal set, generalizing Perron-Frobenius theorem.

Contribution

It develops a framework for analyzing Markov chains with uncertain probabilities using lower and upper expectations, extending classical results to imprecise models.

Findings

Convergence of credal sets to a unique invariant set

Efficient analysis via lower and upper expectations

Generalization of Perron-Frobenius theorem

Abstract

When the initial and transition probabilities of a finite Markov chain in discrete time are not well known, we should perform a sensitivity analysis. This can be done by considering as basic uncertainty models the so-called credal sets that these probabilities are known or believed to belong to, and by allowing the probabilities to vary over such sets. This leads to the definition of an imprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-called lower and upper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at time n evolves as n goes to infinity: under quite unrestrictive conditions, it converges to a uniquely invariant credal set, regardless of the credal set given for the initial state. This leads to a non-trivial generalisation of the classical Perron-Frobenius Theorem to imprecise Markov chains.