Hidden states

Hidden Markov model is basically a Markov chain whose internal state cannot be observed directly but only through some probabilistic function. That is, the internal state of the model only determines the probability distribution of the observed variables.

Let us denote the observations by $\boldsymbol{X}= ( x(1), \ldots, x(T) )$. For each state, the distribution $p(x(t) \vert M_t)$ is defined and independent of the time index $t$. The exact form of this conditional distribution depends on the application. In the simplest case there is only a finite number of different observation symbols. In this case the distribution can be characterised by the point probabilities

$$b_i(m) = p(x(t) = m \vert M_t = i),\quad \forall i,m.$$ (4.4)

Letting $\mathbf{B}= ( b_i(m) )$ and the parameters $\boldsymbol{\theta}= ( \mathbf{A}, \mathbf{B}, \boldsymbol{\pi} )$, the joint probability of an observation sequence and a state sequence can be evaluated by simple extension to Equation (4.3)

$$p(\boldsymbol{X}, \boldsymbol{M}\vert \boldsymbol{\theta}) = \pi_... ...} a_{M_t M_{t+1}} \right] \left[ \prod\limits_{t=1}^{T} b_{M_t} (x(t)) \right].$$ (4.5)

The posterior probability of a state sequence can be derived from this as

$$p(\boldsymbol{M}\vert \boldsymbol{X}, \boldsymbol{\theta}) = \fra... ...1} a_{M_t M_{t+1}} \right] \left[ \prod\limits_{t=1}^{T} b_{M_t} (x(t)) \right]$$ (4.6)

where $P(\boldsymbol{X}\vert \boldsymbol{\theta}) = \sum_{\boldsymbol{M}} P(\boldsymbol{X}, \boldsymbol{M}\vert \boldsymbol{\theta})$.