A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition

IV.A Continuous observation densities in HMMs

1. Why use continuous observation densities, and in what form?
   Quantizing continuous signals (vectors) through a codebook can seriously degrade them. The usual form is a finite mixture per state, \(b_j(O) = \sum_m c_{jm} \mathcal{N}(O; \mu_{jm}, U_{jm})\), which can approximate any finite continuous density arbitrarily closely.
   • Rabiner, 1989, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition (Proceedings of the IEEE), IV.A Continuous observation densities in HMMs
2. How are mixture weights and means re-estimated in a continuous-density HMM?
   Using \(\gamma_t(j, k)\), the probability of being in state \(j\) at time \(t\) with mixture component \(k\) accounting for \(O_t\): \(\bar{c}_{jk}\) = expected times in \(j\) using \(k\) / expected times in \(j\); \(\bar{\mu}_{jk}\) is the \(\gamma\)-weighted average of the observations, and similarly for covariances.
   • Rabiner, 1989, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition (Proceedings of the IEEE), IV.A Continuous observation densities in HMMs