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Lecture 16 Hidden Markov Models

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HMM Until now we only considered IID data. Some data are of sequential nature, i.e. have correlations have time. Example: speech: our mouth produces sounds that represent words. Thus words or syllables are natural hidden states. HMMs are basically MoG models with time structure between the hidden states. There is a probability of making a transition from one hidden state to the next, and there are probabilities for the output variables given the hidden state. The past is independent of the future given the present.

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HMM We like to find answers to the following questions: 1) Can we infer the hidden states given the observed symbols (think speech recognition) 2) Can we forecast new symbols (run in future) 3) Can we learn the parameters of the model 4) Can we compute the probability of an observed sequence? Question 1) is solved by the Viterbi algorithm (it’s sort of k-means like) Question 2 is solved by EM, with the E-step solved by belief propagation and the M-step has analytical updates. Viterbi is much like the E-step since ``max’’ is like ``sum’’

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