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Hidden Markov Model 11/28/07

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Bayes Rule The posterior distribution Select k with the largest posterior distribution. Minimizes the average misclassification rate. Maximum likelihood rule is equivalent to Bayes rule with uniform prior. Decision boundary is

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Naïve Bayes approximation When x is high dimensional, it is difficult to estimate

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Naïve Bayes Classifier When x is high dimensional, it is difficult to estimate But if we assume independence, then it becomes a 1-D problem.

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Naïve Bayes Classifier Usually the independence assumption is not valid. But sometimes the NBC can still be a good classifier. A lot of times simple models may not perform badly.

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Hidden Markov Model

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A coin toss example Scenario: You are betting with your friend using a coin toss. And you see (H, T, T, H, …)

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A coin toss example Scenario: You are betting with your friend using a coin toss. And you see (H, T, T, H, …) But, you friend is cheating. He occasionally switches from a fair coin to a biased coin – of course, the switch is under the table! Fair Biased

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A coin toss example This is what really happening: (H, T, H, T, H, H, H, H, T, H, H, T, …) Of course you can’t see the color. So how can you tell your friend is cheating?

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Hidden Markov Model Hidden state (the coin) Observed variable (H or T)

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Markov Property Hidden state (the coin) Observed variable (H or T)

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Markov Property Fair Biased transition probability prior distribution

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Observation independence Hidden state (the coin) Observed variable (H or T) Emission probability

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Model parameters A = (a ij ) (transition matrix) p(y t | x t ) (emission probability) p(x 1 ) (prior distribution)

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Model inference Infer states when model parameters are known. Both states and model parameters are unknown.

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Viterbi algorithm t-1tt+1 1 2 3 4 state time

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Viterbi algorithm Most probable path: t-1tt+1 1 2 3 4 state time

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Viterbi algorithm Most probable path: t-1tt+1 1 2 3 4 state time

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Viterbi algorithm Most probable path: t-1tt+1 1 2 3 4 state time Therefore, the path can be found iteratively.

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Viterbi algorithm Most probable path: t-1tt+1 1 2 3 4 state time Let v k (i) be the most probable path ending in state k. Then

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Viterbi algorithm Initialization (i=0): Recursion (i=1,...,L): Termination: Traceback (i = L,..., 1):

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Advantage of Viterbi path Identify the most probable path very efficiently. The most probable path is legitimate, i.e., it is realizable by the HMM process.

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Issue with Viterbi path The most probability path does not predict the confidence level of a state estimate. The most probably path may not be much more probable then other paths.

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Posterior distribution Estimate p(x k | y 1,..., y L ). Strategy: This is done by a forward-backward algorithm

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Forward-backward algorithm Estimate f k (i)

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Forward algorithm Estimate f k (i) Initialization: Recursion: Termination:

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Backward algorithm Estimate b k (i)

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Backward algorithm Estimate b k (i) Initialization: Recursion: Termination:

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Probability of fair coin 1 P(fair)

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Probability of fair coin 1 P(fair)

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Posterior distribution Posterior distribution predicts the confidence level of a state estimate. Posterior distribution combines information from all paths. But.. The predicted path may not be legitimate.

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Estimating parameters when state sequence is known Given the state sequence {x k } Define A jk = # transitions from j to k. E k (b) = #emissions of b from k. The maximum likelihood estimates of parameters are:

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Infer hidden states together with model parameters Viterbi training Baum-Welch

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Viterbi training Main idea: Use an iterative procedure Estimate state for fixed parameters using the Viterbi algorithm. Estimate model parameters for fixed states.

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Baum-Welch algorithm Instead of using the Viterbi path to estimate state, consider the expected number of A kl and E k (b)

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Baum-Welch algorithm Instead of using the Viterbi path to estimate state, consider the expected number of A kl and E k (b)

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Baum-Welch is a special case of EM algorithm Given an estimate of parameter t, try to find a better Choose to maximize Q

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Baum-Welch is a special case of EM algorithm E-step: Calculate the Q function M-step: Maximize Q( | t ) with respect to .

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Issue with EM EM only finds local maxima. Solution: –Run multiple EM starting with different initial guesses. –Use more sophisticated algorithm such as MCMC.

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Kelvin Murphy Dynamic Bayesian Network

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Software Kevin Murphy’s Bayes Net Toolbox for Matlab http://www.cs.ubc.ca/~murphyk/Software/ BNT/bnt.html

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Applications (Yi Li) Copy number changes

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Applications Protein-binding sites

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Applications www.biocentral.com Sequence alignment

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Reading list Hastie et al. (2001) the ESL book – p184-185. Durbin et al. (1998) Biological Sequence Analysis –Chapter 3.

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