Definition of Hidden Markov Model

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. .

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