Hidden Markov Models: Basics

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Hidden Markov Models Basics

• Raj Bandyopadhyay
• 11/07/2001

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Our running example

• Hypothetical Nucleic Acid (HNA)
• Two bases or residues H and T
• Assume existence of HNA databases
• Examples
• HTTTHT

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Motivating Example

• HNA sequence database HTT, TTT, HHH, TTH
• 1 2 3
• H T T
• T T T
• H H H
• T T H
• P(H)P(T)0.5 P(H)0.25 P(H)P(T)0.5
• P(T)0.75
• Positions 1 3 are determined by an unbiased
  coin, whereas position 2 is determined by a
  biased coin.

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Motivating Example (contd.)

• Imagine 2 people A and B
• A holds an unbiased coin P(H)P(T)0.5
• B holds a biased coin P(T)0.75, P(H)0.25
• Our HNA database can be explained
• A flips first, followed by B, then A again
• Representation as state diagram (Markov Chain).

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Markov Representation
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Hidden Markov Model

• What if
• The persons flipping the coin are hidden?
• Only the results of the coin flips known?
• I.e. Only emissions known states unknown
• Hidden Markov Model
• What can we infer
• From a given HMM about the data?
• From given data about the generating HMM?

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Terms to understand

• State
• Transition Transition probability
• Emission Emission Probability
• Path

t110.5
t220.5
End
e2H0.25 e2T0.75
e1H0.5 e1T0.5
Start
t1E0.5
t120.5
tS11
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Questions of interest

• For a given HMM
• Probability of generating a particular output
  sequence? Likelihood
• Most probable path? Decoding
• Adjusting probabilities in the light of observed
  sequences? Learning/training

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Likelihood

• Baum-Welch score Likelihood of a sequence s
  Sum of likelihoods of all paths generating s
• -log Lo(M) -ve log likelihood used in practice
• Two kinds of deductions required
• Forward Given observations upto time t, to
  predict sequence state at time t1
• Backward Given observations from time t1, to
  deduce sequence state at time t.

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Calculating Likelihood

• The forward recursive relation
• Dynamic Programming Store previously calculated
  values of at(i) Forward method
• Similarly, the backward recursive relation leads
  to the Backward method

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Example
0.4
0.6
1
End
P(H)0.2 P(T)0.8
P(H)0.3 P(T)0.7
Start
1
1
2
Consider the sequence sHT. Paths generating
s 11, 13 LHT(M) 10.20.40.8
(10.20.6)0.7 a1(1)1 a2(1)10.20.4 a2(2)10.
20.6
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Most Probable Path

• Viterbi Score of HMM M w.r.t sequence O
• Most probable path to generate O
• Recursive method and dynamic programming
  Viterbi Algorithm
• Let pi(t) path ending in state i at time t

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Learning Training HMMs

• Assume we have seen part X of a complete
  sequence Y.
• So far, we have a model M to maximize LX(M)
  likelihood of X given M
• We want a new model M to maximize LY(M), given
  X and M
• This is the Baum-Welch (EM) algorithm

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Expectation-Maximization

• Expectation Step obtain a score for the
  goodness of a new model
• Maximization Step set the new model as that
  with maximum goodness

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New Baum-Welch HMM parameters

• New transition probabilities
• New emission probabilities

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Gradient Descent

• Gradient Descent (Baldi-Chauvin)
• Define log-likelihood as an energy measure
• Derive new model parameters so to minimize this
  energy at every step
• Iterative greedy algorithm
• Advantages over Baum-Welch
• Online updates
• Absorbing 0-probability in Baum-Welch

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A Real Sequence HMM
Matching (main), insert and delete states
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Implementation issues

• Parameter initialization average,
• uniform, random etc.
• Priors initialized to favor transitions towards
  matching (main) states
• Initialization from existing Multiple Alignments
• Model length Average length of input sequences
• Adaptable architecture?????

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HMMs Advantages

• Solid statistical foundation
• Efficient learning algorithms
• Flexible and general model for sequence
  properties
• Unsupervised learning from variable-length, raw
  sequences

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HMMs Disadvantages

• Large number of unstructured parameters
  (emission and transition parameters)
• Need large amounts of data
• Subtle long-range correlations in real sequences
  unaccounted for, due to Markov property