Pattern Classification All materials in these slides were taken from
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Transcript Pattern Classification All materials in these slides were taken from
Pattern
Classification
All materials in these slides were taken
from
Pattern Classification (2nd ed) by R. O.
Duda, P. E. Hart and D. G. Stork, John Wiley
& Sons, 2000
with the permission of the authors and
the publisher
Chapter 3:
Maximum-Likelihood & Bayesian
Parameter Estimation (part 1)
Introduction
Maximum-Likelihood Estimation
Example of a Specific Case
The Gaussian Case: unknown and
Bias
Appendix: ML Problem Statement
2
Introduction
Data availability in a Bayesian framework
We could design an optimal classifier if we knew:
P(i) (priors)
P(x | i) (class-conditional densities)
Unfortunately, we rarely have this complete
information!
Design a classifier from a training sample
No problem with prior estimation
Samples are often too small for class-conditional
estimation (large dimension of feature space!)
Pattern Classification, Chapter1 3
3
A priori information about the problem
Do we know something about the distribution?
find parameters to characterize the distribution
Example: Normality of P(x | i)
P(x | i) ~ N( i, i)
Characterized by 2 parameters
Estimation techniques
Maximum-Likelihood (ML) and the Bayesian estimations
Results are nearly identical, but the approaches are different
Pattern Classification, Chapter1 3
4
Parameters in ML estimation are fixed but
unknown!
Best parameters are obtained by maximizing the
probability of obtaining the samples observed
Bayesian methods view the parameters as
random variables having some known distribution
In either approach, we use P(i | x)
for our classification rule!
Pattern Classification, Chapter1 3
Maximum-Likelihood Estimation
5
Has good convergence properties as the
sample size increases
Simpler than any other alternative techniques
General principle
Assume we have c classes and
P(x | j) ~ N( j, j)
P(x | j) P (x | j, j) where:
( j , j )
1 2
11 22
m
n
( j , j ,..., j , j , cov(x j , x j )...)
Pattern Classification, Chapter2 3
6
Use the information
provided by the training samples to estimate
= (1, 2, …, c), each i (i = 1, 2, …, c) is associated
with each category
Suppose that D contains n samples, x1, x2,…, xn
k n
P(D | ) P(x k | ) F()
k 1
P(D | ) is called the likelihoodof w.r.t. the set of samples)
ML estimate of is, by definition the value that ̂
maximizes P(D | )
“It is the value of that best agrees with the actually
observed training sample”
Pattern Classification, Chapter2 3
7
Pattern Classification, Chapter2 3
8
Optimal estimation
Let = (1, 2, …, p)t and let be the gradient operator
θ
,
,...,
p
1 2
t
We define l() as the log-likelihood function
l() = ln P(D | )
(recall D is the training data)
New problem statement:
determine that maximizes the log-likelihood
θˆ arg max l (θ)
θ
Pattern Classification, Chapter2 3
9
The definition of l() is:
n
l (θ) ln p(x k | θ)
k 1
and
k n
( θl θ ln P(x k | θ))
(eq 6)
k 1
Set of necessary conditions for an optimum is:
l = 0 (eq. 7)
Pattern Classification, Chapter2 3
Example, the Gaussian case: unknown
We assume we know the covariance
p(xi | ) ~ N(, )
(Samples are drawn from a multivariate normal
population)
10
1
1
d
ln p (x k | μ) ln (2 ) (x k μ) t Σ 1 x k μ
2
2
and μ ln p (x k | μ) Σ 1 x k μ (eq. 9)
= therefore:
The ML estimate for must satisfy:
n
1
Σ
(x k μˆ ) 0 from eqs 6,7 & 9
k 1
Pattern Classification, Chapter2 3
11
•
Multiplying by and rearranging, we
obtain:
1 k n
μˆ x k
n k 1
Just the arithmetic average of the samples
of the training samples!
Conclusion:
If P(xk | j) (j = 1, 2, …, c) is supposed to be Gaussian in a ddimensional feature space; then we can estimate the vector
= (1, 2, …, c)t and perform an optimal classification!
Pattern Classification, Chapter2 3
12
Example, Gaussian Case: unknown and
First consider univariate case: unknown and
= (1, 2) = (, 2)
l ln p ( xk | θ)
1
1
ln 22
( xk 1 ) 2
2
2 2
(ln P ( xk | θ))
1
0
θl
(ln P ( xk | θ))
2
1
( xk 1 ) 0
2
2
1 ( xk 1 ) 0
2
2
2
2
2
Pattern Classification, Chapter2 3
13
Summation (over the training set):
n 1
ˆ)0
(
x
ˆ k 1
k 1 2
n
2
n
ˆ
( xk 1 )
1
0
2
k 1 ˆ k 1
ˆ
2
2
(1)
(2)
Combining (1) and (2), one obtains:
1 n
ˆ xk
n k 1
;
n
1
2
2
ˆ
ˆ
( xk )
n k 1
Pattern Classification, Chapter2 3
14
The ML estimates for the multivariate case is
similar
The scalars c and are replaced with vectors
The variance 2 is replaced by the covariance matrix
n
1
μˆ x k
n k 1
n
ˆΣ 1 (x μˆ )( x μˆ ) t
k
k
n k 1
Pattern Classification, Chapter 3
15
Bias
ML estimate for 2 is biased
n 1 2
1 n
2
E ( xi x )
2
n
n i 1
Extreme case: n=1, E[ ] = 0 ≠ 2
As the n increases the bias is reduced
this type of estimator is called asymptotically
unbiased
Pattern Classification, Chapter2 3
16
An elementary unbiased estimator for is:
1 n
t
C
(x k μˆ )( x k μˆ )
n1
k 1
Sample covariance matrix
This estimator is unbiased for all distributions
Such estimators are called absolutely
unbiased
Pattern Classification, Chapter2 3
17
Our earlier estimator for is biased:
n
1
ˆΣ (x μˆ )( x μˆ ) t
k
k
n k 1
In fact it is asymptotically unbiased:
Observe that
n
1
ˆ
C
n
Pattern Classification, Chapter2 3
18
Appendix: ML Problem Statement
Let D = {x1, x2, …, xn}
P(x1,…, xn | ) = 1,nP(xk | ); |D| = n
Our goal is to determine ̂ (value of that
maximizes the likelihood of this sample set!)
Pattern Classification, Chapter2 3
19
|D| = n
x1
. . .x
.x . .
2
n
N(j, j) = P(xj, 1)
P(xj | 1)
P(xj | k)
D1
x10 x11
. ..
x
.
20
Dk
x8
.
.
x
x
. .
1
9
Dc
.
. . .
Pattern Classification, Chapter2 3
20
= (1, 2, …, c)
Problem: find ̂ such that:
Max P(D | ) MaxP (x 1,...,x n | )
n
Max P(x k | )
k 1
Pattern Classification, Chapter2 3