pattern classification. chapter 2 (part 1): bayesian decision theory (sections 2.1-2.2) introduction...
TRANSCRIPT
Pattern Classification
Chapter 2 (Part 1): Bayesian Decision Theory
(Sections 2.1-2.2)
• Introduction
• Bayesian Decision Theory–Continuous Features
Pattern Classification, Chapter 2 (Part 1)
3
Introduction
• The sea bass/salmon example
• State of nature, prior
• State of nature is a random variable
• The catch of salmon and sea bass is equiprobable
• P(1) = P(2) (uniform priors)
• P(1) + P( 2) = 1 (exclusivity and exhaustivity)
Pattern Classification, Chapter 2 (Part 1)
4
• Decision rule with only the prior information
• Decide 1 if P(1) > P(2) otherwise decide 2
• Use of the class–conditional information
• P(x | 1) and P(x | 2) describe the difference in lightness between populations of sea-bass and salmon
Pattern Classification, Chapter 2 (Part 1)
5
Pattern Classification, Chapter 2 (Part 1)
6
• Posterior, likelihood, evidence
• P(j | x) = (P(x | j) * P (j)) / P(x) (BAYES RULE)
• Posterior = (Likelihood * Prior) / Evidence
• Where in case of two categories
2
1
)()|()(j
jjj PxPxP
Pattern Classification, Chapter 2 (Part 1)
7
Pattern Classification, Chapter 2 (Part 1)
8
• Intuitive decision rule given the posterior probabilities:
Given x:
if P(1 | x) > P(2 | x) True state of nature = 1
if P(1 | x) < P(2 | x) True state of nature = 2
Why do this?: Whenever we observe a particular x, the probability of error is :
P(error | x) = P(1 | x) if we decide 2
P(error | x) = P(2 | x) if we decide 1
Pattern Classification, Chapter 2 (Part 1)
9
• Minimizing the probability of error
• Decide 1 if P(1 | x) > P(2 | x); otherwise decide 2
Therefore:
P(error | x) = min [P(1 | x), P(2 | x)]
(Bayes decision)
Pattern Classification, Chapter 2 (Part 1)
10Bayesian Decision Theory – Continuous Features
Generalization of the preceding ideas
• Use of more than one feature
• Use more than two states of nature
• Allowing actions and not only decide on the state of nature
• Introduce a loss of function which is more general than the probability of error
• Allowing actions other than classification primarily allows the possibility of rejection
• Refusing to make a decision in close or bad cases!
• Letting loss function state how costly each action taken is
Pattern Classification, Chapter 2 (Part 1)
11
• Let {1, 2,…, c} be the set of c states of nature (or “classes”)
• Let {1, 2,…, a} be the set of possible actions
• Let (i | j) be the loss for action i when the state of nature is j
Bayesian Decision Theory – Continuous Features
Pattern Classification, Chapter 2 (Part 1)
12
What is the Expected Loss for action i ?
Conditional risk = Expected Loss
cj
1jjjii )x|(P)|()x|(R
For any given x the expected loss is
Pattern Classification, Chapter 2 (Part 1)
13
Overall risk
R = Sum of all R(i | x) for i = 1,…,a
Minimizing R Minimizing R(i | x) for i = 1,…, a
for i = 1,…,a
Conditional risk
cj
1jjjii )x|(P)|()x|(R
Pattern Classification, Chapter 2 (Part 1)
14
Select the action i for which R(i | x) is minimum
R is minimum and R in this case is called the Bayes risk = best performance that can be achieved!
Pattern Classification, Chapter 2 (Part 1)
15Two-Category Classification
1 : deciding 1
2 : deciding 2
ij = (i | j)
loss incurred for deciding i when the true state of nature is j
Conditional risk:
R(1 | x) = 11P(1 | x) + 12P(2 | x)
R(2 | x) = 21P(1 | x) + 22P(2 | x)
Pattern Classification, Chapter 2 (Part 1)
16
Our rule is the following:
if R(1 | x) < R(2 | x)
action 1: “decide 1” is taken
This results in the equivalent rule :
decide 1 if:
(21- 11) P(x | 1) P(1) >
(12- 22) P(x | 2) P(2)
and decide 2 otherwise
Pattern Classification, Chapter 2 (Part 1)
17
x (21- 11)
x (12- 22)
Pattern Classification, Chapter 2 (Part 1)
18Two-Category Decision Theory: Chopping Machine
1 = chop
2 = DO NOT chop
1 = NO hand in machine
2 = hand in machine
11 = (1 | 1) = $ 0.00
12 = (1 | 2) = $ 100.00
21 = (2 | 1) = $ 0.01
22 = (1 | 1) = $ 0.01
Therefore our rule becomes
(21- 11) P(x | 1) P(1) > (12- 22) P(x | 2) P(2)
0.01 P(x | 1) P(1) > 99.99 P(x | 2) P(2)
Pattern Classification, Chapter 2 (Part 1)
19
Our rule is the following:
if R(1 | x) < R(2 | x)
action 1: “decide 1” is taken
This results in the equivalent rule :
decide 1 if:
(21- 11) P(x | 1) P(1) >
(12- 22) P(x | 2) P(2)
and decide 2 otherwise
Pattern Classification, Chapter 2 (Part 1)
20
x 0.01
x 99.99
1 = chop 2 = DO NOT chop
1 = NO hand in machine2 = hand in machine
Pattern Classification, Chapter 2 (Part 1)
21
Exercise
Select the optimal decision where:
= {1, 2}
P(x | 1) N(2, 0.5) (Normal distribution)
P(x | 2) N(1.5, 0.2)
P(1) = 2/3
P(2) = 1/3
43
21
Chapter 2 (Part 2): Bayesian Decision Theory
(Sections 2.3-2.5)
• Minimum-Error-Rate Classification
• Classifiers, Discriminant Functions and Decision Surfaces
• The Normal Density
Pattern Classification, Chapter 2 (Part 1)
23
Minimum-Error-Rate Classification
• Actions are decisions on classesIf action i is taken and the true state of nature is j then
the decision is correct if i = j and in error if i j
• Seek a decision rule that minimizes the probability of error which is the error rate
Pattern Classification, Chapter 2 (Part 1)
24
• Introduction of the zero-one loss function:
Therefore, the conditional risk for each action is:
“The risk corresponding to this loss function is the average (or expected) probability error”
c,...,1j,i ji 1
ji 0),( ji
ijij
cj
jjjii
xPxP
xPxR
)|(1)|(
)|()|()|(1
Average Prob. of Error
Pattern Classification, Chapter 2 (Part 1)
25
• Minimize the risk requires maximize P(i | x)
(since R(i | x) = 1 – P(i | x))
• For Minimum error rate
• Decide i if P (i | x) > P(j | x) j i
Pattern Classification, Chapter 2 (Part 1)
26Two-Category Classification
1 : deciding 1
2 : deciding 2
ij = (i | j)
loss incurred for deciding i when the true state of nature is j
Conditional risk:
R(1 | x) = 11P(1 | x) + 12P(2 | x)
R(2 | x) = 21P(1 | x) + 22P(2 | x)
Pattern Classification, Chapter 2 (Part 1)
27
Our rule is the following:
if R(1 | x) < R(2 | x)
action 1: “decide 1” is taken
This results in the equivalent rule :
decide 1 if:
(21- 11) P(x | 1) P(1) >
(12- 22) P(x | 2) P(2)
and decide 2 otherwise
Pattern Classification, Chapter 2 (Part 1)
28
Likelihood ratio:
The preceding rule is equivalent to the following rule:
Then take action 1 (decide 1)
Otherwise take action 2 (decide 2)
)(
)(.
)|()|(
1
2
1121
2212
2
1
P
PxPxP
if
Pattern Classification, Chapter 2 (Part 1)
29
• Regions of decision and zero-one loss function, therefore:
• If is the zero-one loss function which means:
b1
2
a1
2
)(P
)(P2 then
0 1
2 0 if
)(P
)(P then
0 1
1 0
)|x(P
)|x(P :if decide then
)(P
)(P. Let
2
11
1
2
1121
2212
Pattern Classification, Chapter 2 (Part 1)
30
Pattern Classification, Chapter 2 (Part 1)
31Classifiers, Discriminant Functionsand Decision Surfaces
• The multi-category case
• Set of discriminant functions gi(x), i = 1,…, c
• The classifier assigns a feature vector x to class i
if:
gi(x) > gj(x) j i
Pattern Classification, Chapter 2 (Part 1)
32
Pattern Classification, Chapter 2 (Part 1)
33
• Let gi(x) = - R(i | x)
(max. discriminant corresponds to min. risk!)
• For the minimum error rate, we take gi(x) = P(i | x)
(max. discrimination corresponds to max. posterior!)
gi(x) P(x | i) P(i)
gi(x) = ln P(x | i) + ln P(i)
(ln: natural logarithm!)
Pattern Classification, Chapter 2 (Part 1)
34
• Feature space divided into c decision regions
if gi(x) > gj(x) j i then x is in Ri
(Ri means assign x to i)
• The two-category case• A classifier is a “dichotomizer” that has two discriminant
functions g1 and g2
Let g(x) g1(x) – g2(x)
Decide 1 if g(x) > 0 ; Otherwise decide 2
Pattern Classification, Chapter 2 (Part 1)
35
• The computation of g(x)
)(P
)(Pln
)|x(P
)|x(Pln
)x|(P)x|(P)x(g
2
1
2
1
21
Pattern Classification, Chapter 2 (Part 1)
36
On to higher dimensions!
Pattern Classification, Chapter 2 (Part 1)
37
The Normal Density• Univariate density
• Density which is analytically tractable
• Continuous density
• A lot of processes are asymptotically Gaussian
• Handwritten characters, speech sounds are ideal or prototype corrupted by random process (central limit theorem)
Where: = mean (or expected value) of x 2 = expected squared deviation or variance
,x
2
1exp
2
1)x(P
2
Pattern Classification, Chapter 2 (Part 1)
38
Pattern Classification, Chapter 2 (Part 1)
39
• Multivariate density
• Multivariate normal density in d dimensions is:
where:
x = (x1, x2, …, xd)t (t stands for the transpose vector form)
= (1, 2, …, d)t mean vector = d*d covariance matrix
|| and -1 are determinant and inverse respectively
)x()x(
2
1exp
)2(
1)x(P 1t
2/12/d
Chapter 2 (part 3)Bayesian Decision Theory (Sections
2-6,2-9)
• Discriminant Functions for the Normal Density
• Bayes Decision Theory – Discrete Features
Pattern Classification, Chapter 2 (Part 1)
41Discriminant Functions for the Normal Density
• We saw that the minimum error-rate classification can be achieved by the discriminant function
gi(x) = ln P(x | i) + ln P(i)
• Case of multivariate normal
)(lnln2
12ln
2)()(
2
1)( 1
iii
it
ii Pd
xxxg
Pattern Classification, Chapter 2 (Part 1)
42
Case i = 2I (I stands for the identity matrix)
)category! ththe for thresholdthe called is (
)(Pln2
1w ;w
:where
function) ntdiscrimina (linear wxw)x(g
0i
iiti20i2
ii
0itii
i
Pattern Classification, Chapter 2 (Part 1)
43
• A classifier that uses linear discriminant functions is called “a linear machine”
• The decision surfaces for a linear machine are pieces of hyperplanes defined by:
gi(x) = gj(x)
Pattern Classification, Chapter 2 (Part 1)
44
Pattern Classification, Chapter 2 (Part 1)
45
The hyperplane separating Ri and Rj are given by
always orthogonal to the line linking the means!
)()(
)(ln)(
2
1
0)(
2
2
0
0
jij
i
ji
ji
ji
t
P
P
x
w
xxw
)(2
1)()( 0 jiji PPif x then
Pattern Classification, Chapter 2 (Part 1)
46
Pattern Classification, Chapter 2 (Part 1)
47
Pattern Classification, Chapter 2 (Part 1)
48
Case i = (covariance of all classes are identical but arbitrary!)
Hyperplane separating Ri and Rj
Here the hyperplane separating Ri and Rj is generally not orthogonal to the line between the means!
)(
)()(
)(/)(ln)(
2
1
)(
0)(
10
1
0
jiji
tji
jiji
ji
t
PPx
w
xxw
Pattern Classification, Chapter 2 (Part 1)
49
Pattern Classification, Chapter 2 (Part 1)
50
Pattern Classification, Chapter 2 (Part 1)
51
Case i = arbitrary
The covariance matrices are different for each category
Here the separating surfaces are Hyperquadrics which are: hyperplanes, pairs of hyperplanes, hyperspheres, hyperellipsoids, hyperparaboloids, hyperhyperboloids)
)(Plnln2
1
2
1 w
w
2
1W
:where
wxwxWx)x(g
iii1
iti0i
i1
ii
1ii
0itii
ti
Pattern Classification, Chapter 2 (Part 1)
52
Pattern Classification, Chapter 2 (Part 1)
53
Pattern Classification, Chapter 2 (Part 1)
54
Bayes Decision Theory – Discrete Features
• Components of x are binary or integer valued, x can take only one of m discrete values
v1, v2, …, vm
• Case of independent binary features in 2 category problem
Let x = [x1, x2, …, xd ]t where each xi is either 0 or 1, with probabilities:
pi = P(xi = 1 | 1)
qi = P(xi = 1 | 2)
Pattern Classification, Chapter 2 (Part 1)
55
• The discriminant function in this case is:
0g(x) if and0 g(x) if decide
)(P
)(Pln
q1
p1lnw
:and
d,...,1i )p1(q
)q1(plnw
:where
wxw)x(g
21
2
1d
1i i
i0
ii
iii
0i
d
1ii