The Hidden Markov Model is a finite set of states, each of which is
associated with a (generally multidimensional) probability distribution
. Transitions among the states are governed by a set of probabilities
called transition probabilities. In a particular state an
outcome or observation can be generated, according to the
associated probability distribution. It is only the outcome, not the
state visible to an external observer and therefore states are
``hidden'' to the outside; hence the name Hidden Markov Model.
In order to define an HMM completely, following elements are needed.
denotes the current state.
Transition probabilities should satisfy the normal stochastic constraints,
where denotes the observation symbol in the
alphabet, and the current parameter vector.
Following stochastic constraints must be satisfied.
If the observations are continuous then we will have to use a continuous probability density function, instead of a set of discrete probabilities. In this case we specify the parameters of the probability density function. Usually the probability density is approximated by a weighted sum of M Gaussian distributions ,
should satisfy the stochastic constrains,
Therefore we can use the compact notation
to denote an HMM with discrete probability distributions, while
to denote one with continuous densities. .