lecture_10

Report
ECE 8527
to Machine Learning and Pattern Recognition
8443 – Introduction
Pattern Recognition
LECTURE 10: EXPECTATION MAXIMIZATION (EM)
• Objectives:
Jensen’s Inequality (Special Case)
EM Theorem Proof
EM Example – Missing Data
Application: Hidden Markov Models
• Resources:
Wiki: EM History
T.D.: Brown CS Tutorial
UIUC: Tutorial
F.J.: Statistical Methods
The Expectation Maximization Algorithm (Preview)
ECE 8527: Lecture 10, Slide 1
The Expectation Maximization Algorithm (Cont.)
ECE 8527: Lecture 10, Slide 2
The Expectation Maximization Algorithm
ECE 8527: Lecture 10, Slide 3
Synopsis
• Expectation maximization (EM) is an approach that is used in many ways to
find maximum likelihood estimates of parameters in probabilistic models.
• EM is an iterative optimization method to estimate some unknown parameters
given measurement data. Used in a variety of contexts to estimate missing
data or discover hidden variables.
• The intuition behind EM is an old one: alternate between estimating the
unknowns and the hidden variables. This idea has been around for a long
time. However, in 1977, Dempster, et al., proved convergence and explained
the relationship to maximum likelihood estimation.
• EM alternates between performing an expectation (E) step, which computes
an expectation of the likelihood by including the latent variables as if they
were observed, and a maximization (M) step, which computes the maximum
likelihood estimates of the parameters by maximizing the expected likelihood
found on the E step. The parameters found on the M step are then used to
begin another E step, and the process is repeated.
• This approach is the cornerstone of important algorithms such as hidden
Markov modeling and discriminative training, and has been applied to fields
including human language technology and image processing.
ECE 8527: Lecture 10, Slide 4
Special Case of Jensen’s Inequality
Lemma: If p(x) and q(x) are two discrete probability distributions, then:
 p( x) log p( x)   p( x) log q( x)
x
x
with equality if and only if p(x) = q(x) for all x.
Proof:
 p( x) log p( x)   p( x) log q( x)  0
x
x
 p( x) log p( x)  q( x)  0
x
 p( x) 

  0
p
(
x
)
log
x
 q( x) 
q( x)
p
(
x
)
log
0
x
p( x)
q( x)
q( x)
p
(
x
)
log

p
(
x
)(
 1)
x
x
p( x)
p( x)
The last step follows using a bound for the natural logarithm: lnx   x  1.
ECE 8527: Lecture 10, Slide 5
Special Case of Jensen’s Inequality
Continuing in efforts to simplify:
 p( x) log
x
 q ( x) 
q ( x)
q ( x)
   p( x)   q( x)   p( x)  0..
  p( x)(
 1)  p( x)
p ( x)
p ( x)
x
x
x
x
 p ( x)  x
We note that since both of these functions are probability distributions, they
must sum to 1.0. Therefore, the inequality holds.
The general form of Jensen’s inequality relates a convex function of an integral
to the integral of the convex function and is used extensively in information
theory.
ECE 8527: Lecture 10, Slide 6
The EM Theorem
Theorem: If  P  t y log P t y    P  t y log P  t y  then P  y   P   y  .
t
t
Proof: Let y denote observable data. Let P   y  be the probability distribution
of y under some model whose parameters are denoted by  .
Let P  y  be the corresponding distribution under a different setting  .
Our goal is to prove that y is more likely under  than   .
Let t denote some hidden, or latent, parameters that are governed by the
values of  . Because P  t y  is a probability distribution that sums to 1, we
can write:
log P  y   log P   y    P  t y log P  y   P  t y log P   y 
t
t
Because we can exploit the dependence of y on t and using well-known
properties of a conditional probability distribution.
ECE 8527: Lecture 10, Slide 7
Proof Of The EM Theorem
We can multiply each term by “1”:


P t , y  
P t , y  
  P  t y log  P   y   

log P  y   log P   y    P  t y log  P  y  




P
t
,
y
P
t
,
y
t

 t




 P t , y  
 P t , y  
   P  t y log   

  P  t y log  



t
 P t y   t
 P  t y  
  P  t y log P t , y   P  t y log P  t , y 
t
t
  P  t y log P  t y   P  t y log P t , y 
t
t
  P  t y log P t , y   P  t y log P  t , y 
t
t
where the inequality follows from our lemma.
Explanation: What exactly have we shown? If the last quantity is greater than
zero, then the new model will be better than the old model. This suggests a
strategy for finding the new parameters, θ: choose them to make the last
quantity positive!
ECE 8527: Lecture 10, Slide 8
Discussion
• If we start with the parameter setting  , and find a parameter setting  for
which our inequality holds, then the observed data, y, will be more probable
under  than  .
• The name Expectation Maximization comes about because we take the
expectation of P t , y  with respect to the old distribution P  t , y  and then
maximize the expectation as a function of the argument  .
• Critical to the success of the algorithm is the choice of the proper
intermediate variable, t, that will allow finding the maximum of the expectation
of  P  t y log P t y  .
t
• Perhaps the most prominent use of the EM algorithm in pattern recognition is
to derive the Baum-Welch reestimation equations for a hidden Markov model.
• Many other reestimation algorithms have been derived using this approach.
ECE 8527: Lecture 10, Slide 9
Example: Estimating Missing Data
0 1 2 * 
• Consider a data set with a missing element: D  x1, x 2 , x3 , x 4   ,  ,  ,   
2 0 2 2 
• Let us estimate the value of the missing point assuming a Gaussian model

with a diagonal covariance and arbitrary means: θT  1 2 12  22

• Expectation step:  P  t y log P t y 
t

 3

Q(θ; θ)     ln  p (x k θ   ln  p (x 4 θ  p ( x41 θ 0 ; x42  4)dx41

   k 1



3


   ln  p (x k θ    ln
 k 1
  



 x  
p  41  θ0 

4
  x41  
  
dx
p   θ 
41
   x 

4

    p  41  θ0 dx 
 
 41
4



 

Assuming normal distributions as initial conditions, this can be simplified to:
3
1  12
k 1
2 12
Q (θ; θ)   ln  p (x k θ  
ECE 8527: Lecture 10, Slide 10

(4  2 )2
2 22
 ln( 21 2 )
Example: Gaussian Mixtures
• An excellent tutorial on Gaussian mixture estimation can be found at
J. Bilmes, EM Estimation
• An interactive demo showing convergence of the estimate can be found at
I. Dinov, Demonstration
ECE 8527: Lecture 10, Slide 11
Introduction To Hidden Markov Models
ECE 8527: Lecture 10, Slide 12
Introduction To Hidden Markov Models (Cont.)
ECE 8527: Lecture 10, Slide 13
Introduction To Hidden Markov Models (Cont.)
ECE 8527: Lecture 10, Slide 14
Summary
• Expectation Maximization (EM) Algorithm: a generalization of Maximum
Likelihood Estimation (MLE) based on maximization of a posterior that data
was generated by a model. EM is a special case of Jensen’s inequality.
• Jensen’s Inequality: describes a relationship between two probability
distributions in terms of an entropy-like quantity. A key tool in proving that EM
estimation converges.
• The EM Theorem: proved that estimation of a model’s parameters using an
iteration of EM increases the posterior probability that the data was generated
by the model.
• Demonstrated an application of the EM Theorem to the problem of estimating
missing data point.
• Explained how EM can be used to reestimate parameters in a pattern
recognition system.
• Introduced the concept of a hidden Markov model and explained how we will
use EM to estimate the parameters of this model.
ECE 8527: Lecture 10, Slide 15

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