Class%207:%20Hidden%20Markov%20Models

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Class 7Hidden Markov Models
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Sequence Models

• So far we examined several probabilistic models
• sequence models
• These model, however, assumed that positions are
  independent
• This means that the order of elements in the
  sequence did not play a role
• In this class we learn about probabilistic models
  of sequences

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Probability of Sequences

• Fix an alphabet ?
• Let X1,,Xn be a sequence of random variables
  over ?
• We want to model P(X1,,Xn)

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Markov Chains

• Assumption
• Xi1 is independent of the past once we know Xi
• This allows us to write

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Markov Chains (cont)

• Assumption
• P(Xi1Xi) is the same for all i
• Notation P(Xi1b Xia ) Aab
• By specifying the matrix A and initial
  probabilities, we define P(X1,,Xn)
• To avoid the special case of P(X1), we can use a
  special start state, and denote P(X1 a) Asa

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Example CpG islands

• In human genome, CpG dinucleotides are relatively
  rare
• CpG pairs undergo a process called methylation
  that modifies the C nucleotide
• A methylated C can (with relatively high chance)
  mutate to a T
• Promotor regions are CpG rich
• These regions are not methylated, and thus mutate
  less often
• These are called CpG islands

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CpG Islands

• We construct Markov chain for CpG rich and poor
  regions
• Using maximum likelihood estimates from 60K
  nucleotide, we get two models

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Ratio Test for CpG islands

• Given a sequence X1,,Xn we compute the
  likelihood ratio

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Empirical Evalation
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Finding CpG islands

• Simple Minded approach
• Pick a window of size N (N 100, for example)
• Compute log-ratio for the sequence in the window,
  and classify based on that
• Problems
• How do we select N?
• What do we do when the window intersects the
  boundary of a CpG island?

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Alternative Approach

• Build a model that include states and -
  states
• A state remembers last nucleotide and the type
  of region
• A transition from a - state to a describes a
  start of CpG island

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Hidden Markov Models

• Two components
• A Markov chain of hidden states H1,,Hn with L
  values
• P(Hi1k Hil ) Alk
• Observations X1,,Xn
• Assumption Xi depends only on hidden state Hi
• P(Xia Hik ) Bka

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Semantics
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Example Dishonest Casino
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Computing Most Probable Sequence

• Given x1,,xn
• Output h1,,hn such that

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• Idea
• If we know the value of hi, then the most
  probable sequence on i1,,n does not depend on
  observations before time i
• Let Vi(l) be the probability of the best sequence
  h1,,hi such that hi l

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Dynamic Programming Rule

• so

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Viterbi Algorithm

• Set V0(0) 1, V0(l) 0 for l gt 0
• for i 1, , n
• for l 1,,L
• set
• Let hn argmaxl Vn(l)
• for i n-1,,1
• set hi Pi1(hi1)

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Viterbi Algorithm Example
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Computing Probabilities

• Given x1,,xn
• Output P(x1,,xn )
• How do we sum exponential number of hidden
  sequences?

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Forward Algorithm

• Perform dynamic programming on sequences
• Let fi(l) P(x1,,xi,Hil)
• Recursion rule
• Conclusion

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Computing Posteriors

• How do we compute P(Hi x1,,xn) ?

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Backward Algorithm

• Perform dynamic programming on sequences
• Let bi(l) P(xi1,,xnHil)
• Recursion rule
• Conclusion

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Computing Posteriors

• How do we compute P(Hi x1,,xn) ?

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Dishonest Casino (again)

• Computing posterior probabilities for fair at
  each point in a long sequence

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Learning

• Given a sequence x1,,xn, h1,,hn
• How do we learn Akl and Bka ?
• We want to find parameters that maximize the
  likelihood P(x1,,xn, h1,,hn)
• We simply count
• Nkl - number of times hik hi1l
• Nka - number of times hik xi a

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Learning

• Given only sequence x1,,xn
• How do we learn Akl and Bka ?
• We want to find parameters that maximize the
  likelihood P(x1,,xn)
• Problem
• Counts are inaccessible since we do not observe hi

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• If we have Akl and Bka we can compute

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Expected Counts

• We can compute expected number of times hik
  hi1l
• Similarly

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Expectation Maximization (EM)

• Choose Akl and Bka
• E-step
• Compute expected counts ENkl, ENka
• M-Step
• Restimate
• Reiterate

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EM - basic properties

• P(x1,,xn Akl, Bka) ? P(x1,,xn Akl, Bka)
• Likelihood grows in each iteration
• If P(x1,,xn Akl, Bka) P(x1,,xn Akl,
  Bka)then Akl, Bka is a stationary point of the
  likelihood
• either a local maxima, minima, or saddle point

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Complexity of E-step

• Compute forward and backward messages
• Time Space complexity O(nL)
• Accumulate expected counts
• Time complexity O(nL2)
• Space complexity O(L2)

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EM - problems

• Local Maxima
• Learning can get stuck in local maxima
• Sensitive to initialization
• Require some method for escaping such maxima
• Choosing L
• We often do not know how many hidden values we
  should have or can learn