Forward–backward algorithm: Difference between revisions

Revision as of 10:47, 2 May 2012

Revision as of 10:50, 2 May 2012

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==Forward probabilities==

The following description takeswill as its baseuse matrices of probability values rather than probability distributions., although in general the forward-backward algorithm can be applied to continuous as well as discrete probability models.

We transform the probability distributions related to a given [[hidden Markov model]] into matrix notation as follows.

The transition probabilities <math>\mathbf{P}(X_t\mid X_{t-1})</math> of a given random variable <math>X_t</math> representing all possible states in the hidden Markov model will be represented by the matrix <math>\mathbf{T}</math> where the row index, i, will represent the start state and the column index, j, represents the target state. The example below represents a system where the probability of staying in the same state after each step is 70% and the probability of transitioning to the other state is 30%. The transition matrix is then: