HMMs-2 - Hidden Markov Models (Part 2) BMI/CS 576

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Hidden Markov Models (Part 2) BMI/CS 576 www.biostat.wisc.edu/bmi576.html Mark Craven craven@biostat.wisc.edu Fall 2011 Three important questions How likely is a given sequence? What is the most probable “path” for generating a given sequence? How can we learn the HMM parameters given a set of sequences?
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Learning without hidden information learning is simple if we know the correct path for each sequence in our training set estimate parameters by counting the number of times each parameter is used across the training set 5 C A G T 0 2 2 4 4 begin end 0 4 3 2 1 5 Learning with hidden information 5 C A G T 0 begin end 0 4 3 2 1 5 ? ? ? ? if we don’t know the correct path for each sequence in our training set, consider all possible paths for the sequence estimate parameters through a procedure that counts the expected number of times each parameter is used across the training set
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Learning parameters if we know the state path for each training sequence, learning the model parameters is simple – no hidden information during training – count how often each parameter is used – normalize/smooth to get probabilities – process is just like it was for Markov chain models if we don’t know the path for each training sequence, how can we determine the counts? – key insight: estimate the counts by considering every path weighted by its probability Learning parameters: the Baum-Welch algorithm a.k.a the Forward-Backward algorithm an Expectation Maximization (EM) algorithm – EM is a family of algorithms for learning probabilistic models in problems that involve hidden information in this context, the hidden information is the path that best explains each training sequence
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Learning parameters: the Baum-Welch algorithm algorithm sketch: – initialize parameters of model – iterate until convergence • calculate the expected number of times each transition or emission is used • adjust the parameters to maximize the likelihood of these expected values The expectation step we want to know the probability of generating sequence x with the i th symbol being produced by state k (for all x , i and k ) C A G T A 0.4 C 0.1 G 0.2 T 0.3 A 0.4 C 0.1 G 0.1 T 0.4 A 0.2 C 0.3 G 0.3 T 0.2 begin end 0.5 0.5 0.2 0.8 0.4 0.6 0.1 0.9 0.2 0.8 0 5 4 3 2 1 A 0.1 C 0.4 G 0.4 T 0.1
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The expectation step the forward algorithm gives us , the probability of being in state k having observed the first i characters of x ) ( i f k A 0.4 C 0.1 G 0.2 T 0.3 A 0.2 C 0.3 G 0.3 T 0.2 begin end 0.5 0.5 0.2 0.8 0.4
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HMMs-2 - Hidden Markov Models (Part 2) BMI/CS 576

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