Equivalence of Hidden Markov Models with Continuous Observations

TL;DR

This paper proves that it is possible to determine in polynomial time whether two Hidden Markov Models with continuous observation distributions are equivalent, extending the understanding of HMMs beyond finite observation alphabets.

Contribution

It establishes a polynomial-time algorithm for checking the equivalence of continuous-observation Hidden Markov Models, a problem previously unresolved.

Findings

Polynomial-time algorithm for equivalence testing

Extension of HMM theory to continuous observations

Broader applicability of HMM comparison methods

Abstract

We consider Hidden Markov Models that emit sequences of observations that are drawn from continuous distributions. For example, such a model may emit a sequence of numbers, each of which is drawn from a uniform distribution, but the support of the uniform distribution depends on the state of the Hidden Markov Model. Such models generalise the more common version where each observation is drawn from a finite alphabet. We prove that one can determine in polynomial time whether two Hidden Markov Models with continuous observations are equivalent.