Variational Kullback-Leibler divergence for Hidden Markov models

Divergence measures are widely used tools in statistics and pattern recognition. The Kullback-Leibler (KL) divergence between two hidden Markov models (HMMs) would be particularly useful in the fields of speech and image recognition. Whereas the KL divergence is tractable for many distributions, including Gaussians, it is not in general tractable for mixture models or HMMs. Recently, variational approximations have been introduced to efficiently compute the KL divergence and Bhattacharyya divergence between two mixture models, by reducing them to the divergences between the mixture components. Here we generalize these techniques to approach the divergence between HMMs using a recursive backward algorithm. Two such methods are introduced, one of which yields an upper bound on the KL divergence, the other of which yields a recursive closed-form solution. The KL and Bhattacharyya divergences, as well as a weighted edit-distance technique, are evaluated for the task of predicting the confusability of pairs of words.