Equivalence of Hidden Markov Models with Continuous Observations

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.