More on Hidden Markov Models

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More on Hidden Markov Models
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The Noisy Channel Model

• (some slides adapted from slides byJason Eisner,
  Rada Mihalcea, Bonnie Dorr Christof Monz)

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Our Simple Markovian Tagger
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The Hidden Markov Model Tagger
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Parameters of an HMM

• States A set of states Ss1,,sn
• Transition probabilities A a1,1,a1,2,,an,n
  Each ai,j represents the probability of
  transitioning from state si to sj.
• Emission probabilities A set B of functions of
  the form bi(ot) which is the probability of
  observation ot being emitted by si
• Initial state distribution is the
  probability that si is a start state

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Recognition using an HMM
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Noisy Channel in a Picture

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Noisy Channel Model
real language X
noisy channel X ? Y
yucky language Y
want to recover X from Y
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Noisy Channel Model
real language X
correct spelling
typos
noisy channel X ? Y
yucky language Y
misspelling
want to recover X from Y
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Noisy Channel Model
real language X
(lexicon space)
delete spaces
noisy channel X ? Y
yucky language Y
text w/o spaces
want to recover X from Y
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Noisy Channel Model
real language X
(lexicon space)
pronunciation
noisy channel X ? Y
yucky language Y
speech
want to recover X from Y
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Noisy Channel Model
real language X
English
English ?French
noisy channel X ? Y
yucky language Y
French
want to recover X from Y
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Markovian Tagger HMM Noisy Channel!!
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Review First Two Basic HMM Problems

• Problem 1 (Evaluation) Given the observation
  sequence Oo1,,oT and an HMM model
  , how do we compute the probability of O
  given the model?
• Problem 2 (Decoding) Given the observation
  sequence Oo1,,oT and an HMM model
• , how do we find the
  state sequence that best explains the
  observations?

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The Three Basic HMM Problems

• Problem 3 (Learning) How do we adjust the model
  parameters , to maximize
  ?

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Problem 3 Learning

• Up to now weve assumed that we know the
  underlying model
• Often these parameters are estimated on annotated
  training data, but
• Annotation is often difficult and/or expensive
• Training data is different from the current data
• We want to maximize the parameters with respect
  to the current data, i.e., were looking for a
  model , such that

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Problem 3 Learning

• Unfortunately, there is no known way to
  analytically find a global maximum, i.e., a model
  , such that
• But it is possible to find a local maximum
• Given an initial model , we can always find a
  model , such that

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Forward-Backward (Baum-Welch) algorithm

• Key Idea parameter re-estimation by
  hill-climbing
• From an arbitrary initial parameter instantiation
  , the FB algorithm iteratively re-estimates
  the parameters, improving the probability that a
  given observation was generated by

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Parameter Re-estimation

• Three parameters need to be re-estimated
• Initial state distribution
• Transition probabilities ai,j
• Emission probabilities bi(ot)

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Review
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The Trellis
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Forward Probabilities

• What is the probability that, given an HMM ,
  at time t the state is i and the partial
  observation o1 ot has been generated?

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Forward Probabilities
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Forward Algorithm

• Initialization
• Induction
• Termination

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Forward Algorithm Complexity

• Naïve approach takes O(2TNT) computation
• Forward algorithm using dynamic programming takes
  O(N2T) computations

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Backward Probabilities

• What is the probability that given an HMM and
  given the state at time t is i, the partial
  observation ot1 oT is generated?
• Analogous to forward probability, just in the
  other direction

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Backward Probabilities

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Backward Algorithm

• Initialization
• Induction
• Termination

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Re-estimating Transition Probabilities

• Whats the probability of being in state si at
  time t and going to state sj, given the current
  model and parameters?

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Re-estimating Transition Probabilities

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Re-estimating Transition Probabilities

• The intuition behind the re-estimation equation
  for transition probabilities is
• Formally

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Re-estimating Transition Probabilities

• Defining
• As the probability of being in state si, given
  the complete observation O
• We can say

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Re-estimating Initial State Probabilities

• Initial state distribution is the
  probability that si is a start state
• Re-estimation is easy
• Formally

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Re-estimation of Emission Probabilities

• Emission probabilities are re-estimated as
• Formally where
• Note that here is the Kronecker delta
  function and is not related to the in the
  discussion of the Viterbi algorithm!!

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The Updated Model

• Coming from we get to
  
• by the following
  update rules

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Expectation Maximization

• The forward-backward algorithm is an instance of
  the more general EM algorithm
• The E Step Compute the forward and backward
  probabilities for a give model
• The M Step Re-estimate the model parameters