Maximum a Posteriori Estimation for Multivariate Gaussian Mixture Observations of Markov Chains

In this paper, a framework for maximum a posteriori (MAP) estimation of hidden Markov models (HMM) is presented. Three key issues of MAP estimation, namely, the choice of prior distribution family, the specification of the parameters of prior densities, and the evaluation of the MAP estimates, are addressed. Using HMM's with Gaussian mixture state observation densities as an example, it is assumed that the prior densities for the HMM parameters can be adequately represented as a product of Dirichlet and normal-Wishart densities. The classical maximum likelihood estimation algorithms, namely, the forward-backward algorithm and the segmental k-means algorithm, are expanded, and MAP estimation formulas are developed. Prior density estimation issues are discussed for two classes of applications-parameter smoothing and model adaptation-and some experimental results are given illustrating the practical interest of this approach. Because of its adaptive nature, Bayesian learning is shown to serve as a unified approach for a wide range of speech recognition applications.