Hidden Markov Models And The Variants

This chapter builds upon the reviews in the previous chapter on aspects of probability theory and statistics, including random variables and Gaussian mixture models, and extends the reviews to the Markov chain and the hidden Markov sequence or model (HMM). Central to the HMM is the concept of state, which is itself a random variable typically taking discrete values. Extending from a Markov chain to an HMM involves adding uncertainty or a statistical distribution on each of the states in the Markov chain. Hence, an HMM is a doubly-stochastic process, or probabilistic function of a Markov chain. When the state of the Markov sequence or HMM is confined to be discrete and the distributions associated with the HMM states do not overlap, we reduce it to a Markov chain. This chapter covers several key aspects of the HMM, including its parametric characterization, its simulation by random number generators, its likelihood evaluation, its parameter estimation via the EM algorithm, and its state decoding via the Viterbi algorithm or a dynamic programming procedure. We then provide discussions on the use of the HMM as a generative model for speech feature sequences and its use as the basis for speech recognition. Finally, we discuss the limitations of the HMM, leading to its various extended versions, where each state is made associated with a dynamic system or a hidden time-varying trajectory instead of with a temporally independent stationary distribution such as a Gaussian mixture. These variants of the HMM with state-conditioned dynamic systems expressed in the state-space formulation are introduced as a generative counterpart of the recurrent neural networks to be described in detail in Chap. 13.