HMM - Part 2

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HMM - Part 2

• The EM algorithm
• Continuous density HMM

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The EM Algorithm

• EM Expectation Maximization
• Why EM?
• Simple optimization algorithms for likelihood
  functions rely on the intermediate variables,
  called latent dataFor HMM, the state sequence is
  the latent data
• Direct access to the data necessary to estimate
  the parameters is impossible or difficultFor
  HMM, it is almost impossible to estimate (A, B,
  ?) without considering the state sequence
• Two Major Steps
• E step calculate expectation with respect to the
  latent data given the current estimate of the
  parameters and the observations
• M step estimate a new set of parameters
  according to Maximum Likelihood (ML) or Maximum A
  Posteriori (MAP) criteria

ML vs. MAP
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The EM Algorithm (cont.)

• The EM algorithm is important to HMMs and many
  other model learning techniques
• Basic idea
• Assume we have ? and the probability that each
  Qq occurred in the generation of Oo
• i.e., we have in fact observed a complete
  data pair (o,q) with frequency proportional to
  the probability P(Oo,Qq?)
• We then find a new that maximizes
• 
  
• It can be guaranteed that
• EM can discover parameters of model ? to maximize
  the log-likelihood of the incomplete data,
  logP(Oo?), by iteratively maximizing the
  expectation of the log-likelihood of the complete
  data, logP(Oo,Qq?)

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The EM Algorithm (cont.)
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The EM Algorithm (cont.)
1. Jensens inequality If f is a concave
function, and X is a r.v., then Ef(X)
f(EX) 2. log x x-1
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Solution to Problem 3 - The EM Algorithm

• The auxiliary function
• Where and
  can be expressed as

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Solution to Problem 3 - The EM Algorithm (cont.)

• The auxiliary function can be rewritten as

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Solution to Problem 3 - The EM Algorithm (cont.)

• The auxiliary function is separated into three
  independent terms, each respectively corresponds
  to , , and
• Maximization procedure on can be
  done by maximizing the individual terms
  separately subject to probability constraints
• All these terms have the following form

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Solution to Problem 3 - The EM Algorithm (cont.)

• Proof Apply Lagrange Multiplier

Constraint
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Solution to Problem 3 - The EM Algorithm (cont.)
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Solution to Problem 3 - The EM Algorithm (cont.)
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Solution to Problem 3 - The EM Algorithm (cont.)
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Solution to Problem 3 - The EM Algorithm (cont.)

• The new model parameter set
  can be expressed as

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Discrete vs. Continuous Density HMMs

• Two major types of HMMs according to the
  observations
• Discrete and finite observation
• The observations that all distinct states
  generate are finite in number, i.e., Vv1, v2,
  v3, , vM, vk?RL
• In this case, the observation probability
  distribution in state j, Bbj(k), is defined as
  bj(k)P(otvkqtj), 1?k?M, 1?j?Not
  observation at time t, qt state at time t
• ? bj(k) consists of only M probability values
• Continuous and infinite observation
• The observations that all distinct states
  generate are infinite and continuous, i.e., Vv
  v?RL
• In this case, the observation probability
  distribution in state j, Bbj(v), is defined as
  bj(v)f(otvqtj), 1?j?Not observation at
  time t, qt state at time t
• ? bj(v) is a continuous probability density
  function (pdf) and is often a mixture of
  Multivariate Gaussian (Normal) Distributions

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Gaussian Distribution

• A continuous random variable X is said to have a
  Gaussian distribution with mean µand variance
  s2(sgt0) if X has a continuous pdf in the
  following form

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Multivariate Gaussian Distribution

• If X(X1,X2,X3,,XL) is an L-dimensional random
  vector with a multivariate Gaussian distribution
  with mean vector ? and covariance matrix ?, then
  the pdf can be expressed as
• If X1,X2,X3,,XL are independent random
  variables, the covariance matrix is reduced to
  diagonal, i.e.,

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Multivariate Mixture Gaussian Distribution

• An L-dimensional random vector X(X1,X2,X3,,XL)
  is with a multivariate mixture Gaussian
  distribution if
• In CDHMM, bj(v) is a continuous probability
  density function (pdf) and is often a mixture of
  multivariate Gaussian distributions

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Solution to Problem 3 The Intuitive View
(CDHMM)

• Define a new variable ?t(j,k)
• probability of being in state j at time t with
  the k-th mixture component accounting for ot

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Solution to Problem 3 The Intuitive View
(CDHMM) (cont.)

• Re-estimation formulae for
  are

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Solution to Problem 3 - The EM Algorithm(CDHMM)

• Express with respect to each single
  mixture component

K one of the possible mixture component sequence
along with the state sequence Q
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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)

• The auxiliary function can be written as
• Compared to the DHMM case, we need to further
  solve

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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)

• The new model parameter set can be
  derived as

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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)

• The new model parameter sets can
  be derived as

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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)
We thus solve
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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)
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Solution to Problem 3 - The EM Algorithm(CDHMM)
(cont.)
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HMM Topology

• Speech is a time-evolving non-stationary signal
• Each HMM state has the ability to capture some
  quasi-stationary segment in the non-stationary
  speech signal
• A left-to-right topology is a natural candidate
  to model the speech signal
• Each state has a state-dependent output
  probability distribution that can be used to
  interpret the observable speech signal
• It is general to represent a phone using 35
  states (English) and a syllable using 68 states
  (Mandarin Chinese)

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HMM Limitations

• HMMs have proved themselves to be a good model of
  speech variability in time and feature space
  simultaneously
• There are a number of limitations in the
  conventional HMMs
• The state duration follows an exponential
  distribution
• Dont provide adequate representation of the
  temporal structure of speech
• First order (Markov) assumption the state
  transition depends only on the previous state
• Output-independent assumption all observation
  frames are dependent on the state that generated
  them, not on neighboring observation frames
• HMMs are well defined only for processes that are
  a function of a single independent variable, such
  as time or one-dimensional position
• Although speech recognition remains the dominant 
  field in which HMMs are applied, their use has
  been spreading steadily to other fields

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ML vs. MAP

• Estimation principle based on observations
  Oo1, o2, , oT
• The Maximum Likelihood (ML) principlefind the
  model parameter ? so that the likelihood P(O?)
  is maximum
• for example, if ? ?,? is the parameters of a
  multivariate normal distribution, and O is i.i.d.
  (independent, identically distributed), then the
  ML estimate of ? ?,? is
• The Maximum a Posteriori (MAP) principlefind
  the model parameter ? so that the likelihood P(?
  O) is maximum

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A Simple Example
The Forward/Backward Procedure
S1
S1
S1
State
S2
S2
S2
1 2 3 Time
o1
o2
o3
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A Simple Example (cont.)









q 1 1 1
q 1 1 2
Total 8 paths
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A Simple Example (cont.)
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Appendix - Matrix Calculus

• Notation

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Appendix - Matrix Calculus (cont.)

• Property 1
• proof

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Appendix - Matrix Calculus (cont.)

• Property 1 - Extension
• proof

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Appendix - Matrix Calculus (cont.)

• Property 2
• proof

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Appendix - Matrix Calculus (cont.)

• Property 3
• proof

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Appendix - Matrix Calculus (cont.)

• Property 4
• proof

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Appendix - Matrix Calculus (cont.)

• Property 5
• proof

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Appendix - Matrix Calculus (cont.)

• Property 6
• proof

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Appendix - Matrix Calculus (cont.)

• Property 7
• proof

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Appendix - Matrix Calculus (cont.)

• Property 8
• proof

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Appendix - Matrix Calculus (cont.)

• Property 9
• proof

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