Continuous observations

An obvious extension to the basic HMM model is to allow continuous observation space instead of a finite number of discrete symbols. In this model the parameters $\mathbf{B}$ cannot be described as a simple matrix of point probabilities but rather as a complete pdf over the continuous observation space for each state. Therefore the values of $b_i(m)$ in Equation (4.4) must be replaced with a continuous probability distribution

$$b_i(x(t)) = p(x(t) \vert M_t = i), \quad \forall x(t), i.$$ (4.7)

This model is called continuous density hidden Markov model (CDHMM). The probability of an observation sequence evaluated in Equation (4.5) stays the same. The conditional distributions $b_i(x(t))$ can in principle be arbitrary but usually they are restricted to be finite mixtures of simple parametric distributions, like Gaussians [48].