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.

- The number of states of the model,
*N*. - The number of observation symbols in the alphabet,
*M*. If the observations are continuous then*M*is infinite. - A set of state transition probabilities .
where denotes the current state.

Transition probabilities should satisfy the normal stochastic constraints,and

- A probability distribution in each of the
states, .
where denotes the observation symbol in the alphabet, and the current parameter vector.

Following stochastic constraints must be satisfied.and

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 ,where,

should satisfy the stochastic constrains,

and

- The initial state distribution, .

where,

Therefore we can use the compact notation

to denote an HMM with discrete probability distributions, while

to denote one with continuous densities. .

Fri May 10 20:35:10 MET DST 1996