The moments of matched and mismatched hidden Markov models

An algorithm for computing the moments of matched and mismatched hidden Markov models (HMMs) from their defining parameters is presented. The algorithm is an extension of the usual forward-backward linear recursion. The algorithm computes the joint moments of the posterior likelihood functions (i.e. the scores) by a multilinear recursion involving the joint moments of the random variables associated with the hidden states of the Markov chain. Examples comparing the first two theoretical moments to simulation results are presented. They are of independent interest because they indicate that the distribution of the posterior likelihood function scores for mismatched models are asymptotically log-normal in important special cases and, therefore, are characterized asymptotically by the first two moments alone. One example shows that classification using a suboptimal statistic reliably distinguishes between sufficiently long quasi-stationary signals with a reasonable amount of computational effort