Inference and minimization of hidden Markov Chains

Abstract

A hidden Markov chain (hmc) is a finite er-godic Markov chain in which each of the states is labelled 0 or 1. As the Markov chain moves through a random trajectory, the hmc emits a 0 or a 1 at each time step according to the label of the state just entered. The inference problem is to construct a mechanism which will emit O's and l's and which is equivalent to a given hmc in the sense of having identical long-term behavior. We define the inference problem in a learning setting in which an algorithm can query an oracle for the long-term probability of any binary string. We prove that inference is hard: any algorithm for inference must make exponentially many oracle calls. Our method is information-theoretic and does not depend on separation assumptions for any complexity classes. We show that the related discrimination problem is also hard, but that on a nontrivial subclass of hmc's there is a randomized algorithm for discrimination. Finally, we give a polynomial-time algorithm for reducing a hidden Markov chain to its minimal form, and from this there follows a new algorithm for equivalence of hmc's.