Upper and lower bounds for approximation of the Kullback-Leibler divergence between Hidden Markov models

The Kullback-Leibler (KL) divergence is often used for a similarity comparison between two Hidden Markov models (HMMs). However, there is no closed form expression for computing the KL divergence between HMMs, and it can only be approximated. In this paper, we propose two novel methods for approximating the KL divergence between the left-to-right transient HMMs. The first method is a product approximation which can be calculated recursively without introducing extra parameters. The second method is based on the upper and lower bounds of KL divergence, and the mean of these bounds provides an available approximation of the divergence. We demonstrate the effectiveness of the proposed methods through experiments including the deviations to the numerical approximation and the task of predicting the confusability of phone pairs. Experimental results show that the proposed product approximation is comparable with the current variational approximation, and the proposed approximation based on bounds performs better than current methods in the experiments.