Hidden Markov Model

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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!
Biased
Fair
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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 cant 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
transition probability
prior distribution
Biased
Fair
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Observation independence
Hidden state (the coin)
Observed variable (H or T)
Emission probability
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Model parameters

• A (aij) (transition matrix)
• p(yt xt) (emission probability)
• p(x1) (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
time
t-1
t
t1
1
2
state
3
4
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Viterbi algorithm
time

• Most probable path

t-1
t
t1
1
2
state
3
4
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Viterbi algorithm
time

• Most probable path

t-1
t
t1
1
2
state
3
4
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Viterbi algorithm
time

• Most probable path

t-1
t
t1
1
2
state
3
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Therefore, the path can be found iteratively.
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Viterbi algorithm
time

• Most probable path

t-1
t
t1
1
2
state
3
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Let vk(i) be the most probable path ending in
state k. Then
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Viterbi algorithm

• Initialization (i0)
• Recursion (i1,...,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(xk y1, ..., yL).
• Strategy
• This is done by a forward-backward algorithm

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Forward-backward algorithm

• Estimate fk(i)

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Forward algorithm
Estimate fk(i)
Initialization Recursion Termination
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Backward algorithm
Estimate bk(i)
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Backward algorithm
Estimate bk(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 xk
• Define
• Ajk transitions from j to k.
• Ek(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 Akl and
  Ek(b)

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Baum-Welch algorithm

• Instead of using the Viterbi path to estimate
  state, consider the expected number of Akl and
  Ek(b)

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Baum-Welch is a special case of EM algorithm

• Given an estimate of parameter qt , try to find a
  better q.

• Choose q 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(qqt) with respect to q.

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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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Dynamic Bayesian Network
Kelvin Murphy
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Software

• Kevin Murphys Bayes Net Toolbox for Matlab
• http//www.cs.ubc.ca/murphyk/Software/BNT/bnt.ht
  ml

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Applications
Copy number changes
(Yi Li)
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Applications
Protein-binding sites
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Applications
Sequence alignment
www.biocentral.com
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Reading list

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