Linear regression for Hidden Markov Model (HMM) parameters is widely used for the adaptive training of time series pattern analysis especially for speech processing. The regression parameters are usually shared among sets of Gaussians in HMMs where the Gaussian clusters are represented by a tree. This paper realizes a fully Bayesian treatment of linear regression for HMMs considering this regression tree structure by using variational techniques. This paper analytically derives the variational lower bound of the marginalized log-likelihood of the linear regression. By using the variational lower bound as an objective function, we can algorithmically optimize the tree structure and hyper-parameters of the linear regression rather than heuristically tweaking them as tuning parameters. Experiments on large vocabulary continuous speech recognition confirm the generalizability of the proposed approach, especially when the amount of adaptation data is limited.
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