IMPRECISE MARKOV CHAINS AND THEIR LIMIT BEHAVIOR

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-calledcredal setsthat 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 animprecise Markov chain. We show that the time evolution of such a system can be studied very efficiently using so-calledlowerandupper expectations, which are equivalent mathematical representations of credal sets. We also study how the inferred credal set about the state at timenevolves asn→∞: 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 generalization of the classical Perron–Frobenius theorem to imprecise Markov chains.

AN ALGORITHM TO COMPUTE THE UPPER ENTROPY FOR ORDER-2 CAPACITIES

The upper entropy of a credal set is the maximum of the entropies of the probabilities belonging to it. Although there are algorithms for computing the upper entropy for the particular cases of credal sets associated to belief functions and probability intervals, there is none for a more general model. In this paper, we shall present an algorithm to obtain the upper entropy for order-2 capacities. Our algorithm is an extension of the one presented for belief functions, and proofs of correctness are provided. By using a counterexample, we shall also prove that this algorithm is not valid for general lower probabilities as it computes a value which is strictly greater than the maximum of entropy.