Viterbi algorithm for Hidden Markov Models

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Viterbi algorithm forHidden Markov Models
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Hidden Markov Models

• Discrete Markov chain on state space augmented by
  states B and E, with transition probability
  matrix a (aij).
• State at time i, denoted ?i, i 0, , L1, ?0
  B, ?L1 E. States are not observable.
• For each time moment, the process generates
  state-dependent observable emissions xi, i 0,
  , L. Probability of emission is denoted e?(x).

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CpG-island example
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Basic questions for HMMs

• Find the most likely path of states, given
  observations.
• Find the most likely state at a given time
  moment, given observations. To accomplish this,
  it is necessary to find the distribution of
  states at a given time moment, given
  observations.
• Recurrent algorithm to answer these questions is
  dynamic programming (Viterbis version)

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HMM for paired alignments

• Internal states, M
• Emissions, M, X, Y, indistinguishable
• In a way, inversion of the real-life situation
  True alignment produces sequences X and Y.

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Viterbi algorithm yields the following recursions
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Version of the algorithm for VM ln(vM)cM, VX
ln(vX)cX, VY ln(vY)cY
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Indistinguishable from the Needleman and Wunsch
algorithm