Summary of the Hidden Markov Models
(Source: http://www.comp.leeds.ac.uk/roger/HiddenMarkovModels/html_dev/summary/s1_pg1.html)
Summary
Frequently, patterns do not appear in isolation but as part of a series in time  this progression can sometimes be used to assist in their recognition. Assumptions are usually made about the time based process  a common assumption is that the process's state is dependent only on the preceding N states  then we have an order N Markov model. The simplest case is N=1.
Various examples exists where the process states (patterns) are not directly observable, but are indirectly, and probabalistically, observable as another set of patterns  we can then define a hidden Markov model  these models have proved to be of great value in many current areas of research, notably speech recognition.
Such models of real processes pose three problems that are amenable to immediate attack; these are :
 Evaluation : with what probability does a given model generate a given sequence of observations. The forward algorithm solves this problem efficiently.

Decoding : what sequence of hidden (underlying) states most probably generated a given sequence of observations. The Viterbi algorithm solves this problem efficiently.
 Learning : what model most probably underlies a given sample of observation sequences  that is, what are the parameters of such a model. This problem may be solved by using the forwardbackward algorithm.
HMMs have proved to be of great value in analysing real systems; their usual drawback is the oversimplification associated with the Markov assumption  that a state is dependent only on predecessors, and that this dependence is time independent.
A full exposition on HMMs may be found in:
L R Rabiner and B H Juang, `An introduction to HMMs', iEEE ASSP Magazine, 3, 416.