Probabilistic Distances between Finite-State Finite-Alphabet Hidden Markov Models

Abstract

This paper considers the problem of evaluating a probabilistic distance between homogeneous, first-order, finite-state finite-alphabet hidden Markov models (HMMs). Our approach is based on a correspondence between probability measures and HMMs established in this paper. Using a probability measure transformation technique, we obtain recursive expressions for the relative entropy between the marginal probability distributions of two HMMs under consideration. Also, the relative entropy rate, as a time-averaged value of the above relative entropy, is obtained. These expressions are given in terms of the parameters of the given HMMs. Using the change of measure, we show that the probabilistic distance between HMMs considered in the existing literature can be expressed in terms of a conditional expectation given the σ-algebra generated by the observation process. This representation allows us to evaluate this distance using the information state approach.