Hidden Markov Model 11/28/07

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

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

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.

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.

Hidden Markov Model

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

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

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?

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

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

Markov Property Fair Biased transition probability prior distribution

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

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

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

Viterbi algorithm t-1tt state time

Viterbi algorithm Most probable path: t-1tt state time

Viterbi algorithm Most probable path: t-1tt state time

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

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

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

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.

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.

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

Forward-backward algorithm Estimate f k (i)

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

Backward algorithm Estimate b k (i)

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

Probability of fair coin 1 P(fair)

Probability of fair coin 1 P(fair)

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.

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:

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

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

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

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

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

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

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

Kelvin Murphy Dynamic Bayesian Network

Software Kevin Murphy’s Bayes Net Toolbox for Matlab BNT/bnt.html

Applications (Yi Li) Copy number changes

Applications Protein-binding sites

Applications Sequence alignment

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