Hidden Markov Models

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Hidden Markov Models
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K

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Outline

• Hidden Markov Models Formalism
• The Three Basic Problems of HMMs
• Solutions
• Applications of HMMs for Automatic Speech
  Recognition (ASR)

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Example The Dishonest Casino

• A casino has two dice
• Fair die
• P(1) P(2) P(3) P(4) P(5) P(6) 1/6
• Loaded die
• P(1) P(2) P(3) P(4) P(5) 1/10
• P(6) 1/2
• Casino player switches back--forth between fair
  and loaded die once in a while
• Game
• You bet 1
• You roll (always with a fair die)
• Casino player rolls (maybe with fair die, maybe
  with loaded die)
• Highest number wins 2

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Question 1 Evaluation

• GIVEN
• A sequence of rolls by the casino player
• 12455264621461461361366616646616366163661636165
• QUESTION
• How likely is this sequence, given our model of
  how the casino works?
• This is the EVALUATION problem in HMMs

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Question 2 Decoding

• GIVEN
• A sequence of rolls by the casino player
• 12455264621461461361366616646616366163661636165
• QUESTION
• What portion of the sequence was generated with
  the fair die, and what portion with the loaded
  die?
• This is the DECODING question in HMMs

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Question 3 Learning

• GIVEN
• A sequence of rolls by the casino player
• 12455264621461461361366616646616366163661636165
• QUESTION
• How loaded is the loaded die? How fair is the
  fair die? How often does the casino player change
  from fair to loaded, and back?
• This is the LEARNING question in HMMs

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The dishonest casino model
0.05
0.95
0.95
FAIR
LOADED
P(1F) 1/6 P(2F) 1/6 P(3F) 1/6 P(4F)
1/6 P(5F) 1/6 P(6F) 1/6
P(1L) 1/10 P(2L) 1/10 P(3L) 1/10 P(4L)
1/10 P(5L) 1/10 P(6L) 1/2
0.05
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Example the dishonest casino

• Let the sequence of rolls be
• O 1, 2, 1, 5, 6, 2, 1, 6, 2, 4
• Then, what is the likelihood of
• X Fair, Fair, Fair, Fair, Fair, Fair, Fair,
  Fair, Fair, Fair?
• (say initial probs P(t0,Fair) ½,
  P(t0,Loaded) ½)
• ½ ? P(1 Fair) P(Fair Fair) P(2 Fair) P(Fair
  Fair) P(4 Fair)
• ½ ? (1/6)10 ? (0.95)9 .00000000521158647211
  0.5 ? 10-9

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Example the dishonest casino

• So, the likelihood the die is fair in all this
  run
• is just 0.521 ? 10-9
• OK, but what is the likelihood of
• X Loaded, Loaded, Loaded, Loaded, Loaded,
  Loaded, Loaded, Loaded, Loaded, Loaded?
• ½ ? P(1 Loaded) P(Loaded, Loaded) P(4
  Loaded)
• ½ ? (1/10)8 ? (1/2)2 (0.95)9 .000000000787811762
  15 7.9 ? 10-10
• Therefore, it is after all 6.59 times more likely
  that the die is fair all the way, than that it is
  loaded all the way.

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Example the dishonest casino

• Let the sequence of rolls be
• O 1, 6, 6, 5, 6, 2, 6, 6, 3, 6
• Now, what is the likelihood X F, F, , F?
• ½ ? (1/6)10 ? (0.95)9 0.5 ? 10-9, same as
  before
• What is the likelihood
• X L, L, , L?
• ½ ? (1/10)4 ? (1/2)6 (0.95)9 .000000492382351347
  35 0.5 ? 10-7
• So, it is 100 times more likely the die is loaded

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HMM Timeline
time ?

• Arrows indicate probabilistic dependencies.
• xs are hidden states, each dependent only on the
  previous state.
• The Markov assumption holds for the state
  sequence.
• os are observations, dependent only on their
  corresponding hidden state.

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HMM Formalism

• An HMM ? can be specified by 3 matrices P, A,
  B
• P pi are the initial state probabilities
• A aij are the state transition probabilities
  Pr(xjxi)
• B bik are the observation probabilities
  Pr(okxi)

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Generating a sequence by the model

• Given a HMM, we can generate a sequence of length
  n as follows
• Start at state xi according to prob ?i
• Emit letter o1 according to prob bi(o1)
• Go to state xj according to prob aij
• until emitting oT

1
?2
2
2
0
N
b2o1
o1
o2
o3
oT
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The three main questions on HMMs

• Evaluation
• GIVEN a HMM ?, and a sequence O,
• FIND Prob O ?
• Decoding
• GIVEN a HMM ?, and a sequence O,
• FIND the sequence X of states that maximizes
  PX O, ?
• Learning
• GIVEN a sequence O,
• FIND a model ? with parameters ?, A and B
  that
• maximize P O ?

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Problem 1 Evaluation

• Find the likelihood a sequence is generated by
  the model

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Probability of an Observation
o1
ot
ot-1
ot1
Given an observation sequence and a model,
compute the probability of the observation
sequence
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Probability of an Observation
Let X x1 xT be the state sequence.
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Probability of an Observation
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HMM Evaluation (cont.)

• Why isnt it efficient?
• For a given state sequence of length T we have
  about 2T calculations
• Let N be the number of states in the graph.
• There are NT possible state sequences.
• Complexity O(2TNT )
• Can be done more efficiently by the
  forward-backward (F-B) procedure.

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The Forward Procedure (Prefix Probs)
The probability of being in state i after
generating the first t observations.
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Forward Procedure
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Forward Procedure
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Forward Procedure
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Forward Procedure
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Forward Procedure
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Forward Procedure
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Forward Procedure
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Forward Procedure
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The Forward Procedure

• Initialization
• Iteration
• Termination
• Computational Complexity O(N2T)

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Another Version The Backward Procedure (Suffix
Probs)
x1
xt1
xT
xt
xt-1
oT
o1
ot
ot-1
ot1
Probability of the rest of the states given the
first state
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Problem 2 Decoding

• Find the best state sequence

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Decoding

• Given an HMM and a new sequence of observations,
  find the most probable sequence of hidden states
  that generated these observations
• In general, there is an exponential number of
  possible sequences.
• Use dynamic programming to reduce search space to
  O(n2T).

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Viterbi Algorithm
x1
xt-1
j
oT
o1
ot
ot-1
ot1
The state sequence which maximizes the
probability of seeing the observations up to time
t-1, landing in state j, and seeing the
observation at time t.
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Viterbi Algorithm
x1
xt-1
j
oT
o1
ot
ot-1
ot1
Initialization
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Viterbi Algorithm
x1
xt-1
xt
xt1
Recursion
Prob. of ML state
Name of ML state
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Viterbi Algorithm
x1
xt-1
xt
xt1
xT
Termination
Read out the most likely state sequence,
working backwards.
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Viterbi Training

• Initialization Same as Baum-Welch
• Iteration
• Perform Viterbi, to find the optimal state
  sequence.
• Calculate P(i,j) and ?i(t) according to the
  optimal state sequence.
• Calculate the new parameters A, B and ?.
• Until convergence
• Notes
• In general, worse performance than Baum-Welch

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Problem 3 Learning

• Re-estimate the parameters of the model based on
  training data

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Learning by Parameter Estimation

• Goal Given an observation sequence, find the
  model that is most likely to produce that
  sequence.
• Problem We dont know the relative frequencies
  of hidden visited states.
• No analytical solution is known for HMMs.
• We will approach the solution by successive
  approximations.

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The Baum-Welch Algorithm

• Find the expected frequencies of possible values
  of the hidden variables.
• Compute the maximum likelihood distributions of
  the hidden variables (by normalizing, as usual
  for MLE).
• Repeat until convergence.
• This is the Expectation-Maximization (EM)
  algorithm for parameter estimation.
• Applicable to any stochastic process, in theory.
• Special case for HMMs is called the Baum-Welch
  algorithm.

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Arc and State Probabilities
A
A
A
A
B
B
B
B
B
Probability of traversing an arc From state i (at
time t) to state j (at time t1)
Probability of being in state i at time t.
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Aggregation and Normalization
A
A
A
A
B
B
B
B
B
Now we can compute the new MLEs of the model
parameters.
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The Baum-Welch Algorithm

1. Initialize A,B and ? (Pick the best-guess for
   model parameters or arbitrary)
2. Repeat
3. Calculate and
4. Calculate and
5. Estimate , and
6. Until the changes are small enough

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The Baum-Welch Algorithm Comments

• Time Complexity
• iterations ? O(N2T)
• Guaranteed to increase the (log) likelihood of
  the model
• P(? O) P(O, ?) / P(O) P(O ?) P(?) / P(O)
• Not guaranteed to find globally best parameters
• Converges to local optimum, depending on initial
  conditions
• Too many parameters / too large model -
  Overtraining

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Application Automatic Speech Recognition
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Example (1)
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Example (2)
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Example (3)
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Example (4)
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Phones
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Speech Signal

• Waveform
• Spectrogram

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Speech Signal cont.
Articulation
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Feature Extraction
Frame 1
Frame 2
Feature VectorX1
Feature VectorX2
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Simple Features Extraction - LPC

• Speech difference equation for a p-th order
  filter
• Want to minimize the mean squared prediction
  error
• The aks create the feature vector.

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