Hidden Markov Model

From lingwiki

A Hidden Markov Model (HMM) is a kind of statistical model, or probabilistic function of a Markov Chain. These models are commonly associated with the question, "Given an observed output sequence, which state sequence is most likely to have caused it?" Unknown parameters are assumed to exist in the model that can be derived from the observable parameters. HMMs are hence used to model data generation; given some sample data, we want to construct a model that will account for future data whose type is the same of that given in the sample.

Formal Definition

If one defines

1. An output alphabet, Y = {0,1,...,b − 1}
2. A state space, S = {1,2,...,c} with a unique starting space s₀
3. A probability distribution of transitions between states p(s' | s)
4. An output probability distribution q(y | s,s') associated with transitions from state s to state s',

then the probability of observing an HMM output string y₁,y₂,...,y_k is given by

Use in Speech Recognition

HMMs are used in the construction of acoustic models and language models.