Course Calendar Class DATE Contents

1 Sep. 26 Course information & Course overview

2 Oct. 4 Bayes Estimation

3 〃 11 Classical Bayes Estimation - Kalman Filter -

4 〃 18 Simulation-based Bayesian Methods

5 〃 25 Modern Bayesian Estimation ：Particle Filter

6 Nov. 1 HMM(Hidden Markov Model)

Nov. 8 No Class

7 〃 15 Bayesian Decision

8 〃 29 Non parametric Approaches

9 Dec. 6 PCA(Principal Component Analysis)

10 〃 13 ICA(Independent Component Analysis)

11 〃 20 Applications of PCA and ICA

12 〃 27 Clustering, k-means et al.

13 Jan. 17 Other Topics 1 Kernel machine.

14 〃 22(Tue) Other Topics 2

Lecture Plan

Bayes Decision

1. Introduction 1.1 Pattern Recognition- 1.2 An Example Classification/Decision Theory 2. Bayes Decision Theory 2.1 Decision using Posterior Probability 2.2 Decision by Minimizing Risk 3. Discriminate Function 4. Gaussian Case

1. Introduction

3

1.1 Pattern Recognition The second part of this course is concerned about Pattern Recognition.

Pattern recognitions (Machine Learning) want to give very high skills

for sensing and taking actions as humans do according to what they

observe.

Definitions of Pattern Recognition appeared in books

“The assignment of a physical object or event to one of several pre-

specified categories”

by Duda et al.[1]

“The science that concerns the description or classification

(recognition) of measurements”

by Schalkoff (Wiley Online Library)

Fish-Sorting Process

Sea bass 鱸

Salmon 鮭

R.O. Duda, P.E. Hart, and D. G. Stork, “Pattern Classification”, John Wiley & Sons, 2nd edition, 2004

1

2

.

:: feature vector in 2-d feature space

:

: action

"Correct dicision " should be an appropriate function of data

eg

x lightness

x width

x

x

x

x

1.2 An Example (Duda, Hart, & Stork 2004)

5

Automatic Fish-Sorting Process

action １

belt conveyer action 2

Typical pattern Recognition issues:

■ Classification ■ Regression

■ Clustering ■ Dimension Reduction

(Visualization)

Pattern Recognition System

data

Measurement Preprocessing

Dimension Reduction Feature Selection

Recognition Classification

Model change Evaluation

analysis results

PCA (ICA)

Clustering Cross-Validation PDF estimation

PDF: Probability Density Function

7

Classification/ Decision Theory

Suppose we observe fish image data x, then we want to classify it to

“sea bass” or “salmon” based on the joint probability distributions

The classification problem is to answer “How do we make the best

decision?”

p ," sea bass" , p ," salmon"x x

x1

x2

Decision Boundary

Classification:

Assign input vector to

one of two classes

R2 R1

8

Framework: - Two Category case (fish sorting example) -

■ State of nature (Class) ω (discrete random variable)

■ Prior Probability

■ Class-conditioned Probability (Likelihood)

Measurement x : brightness of fish (scalar continuous variable)

Class-conditional probability density function for each class:

1

2

: sea bass

: salmon

2. Bayes’ Decision Theory

1 2

1 2

,

where 1

P P

P P

1 1

2 2

PDF for given that the state of nature is

PDF for given that the state of nature is

p x x

p x x

9

Fig. 1 Class-conditioned probabilities

10

2.1 Decision Using Posterior Probability

■ Posterior Probabilities

■ Decision Rule (1) Minimizing error probability

■ Decision Rule (2) Likelihood ratio

the probability of being given that has been measuredDefine

Bayes rule derives

j

j j

j

xjP x

p x PP x

p x

1 21

2 1 2

if >

if <

P x P x

P x P x

Decide

11 1

2 22

if p x P

Pp x

Decide

independent of

observation x

(1)

(2)

(3)

11 Fig. 2 Decision

(a) Posterior Probabilities

(b) Likelihood ratio

12

Probability of Error

■ Error probability for a measurement x by decision

■ Average probability of error

2 1 2 2 1 1

1 2 1 2

if we decide ( )2 1 1

if we decide ( )1 2 2

:

P x P x P x P P x P

P x x R

P x x R

P error xEx

p x dx p x dx dx dx

P error x p x dx

P error x

P error

R R R R

(4)

(5)

Fig. 3 P(error)

13

2.2 Decision by Minimizing Risk

■ Alternate Bayes Decision based on risk which defines “how much

costly each action is ?”

Suppose we observe x then take action according to make a decision

(ωi) if the true state of nature is ωj , we introduce the loss function

■ Example of loss function

From a medical image we want to classify (determine) whether it

contains cancer tissues or not.

i j

i

1 2

1 2

cancer, normal,

cancer, normal

cancer normal

cancer 0 1

normal 100 0

i j

1

2

1 2

(6)

Loss Function

Expected Loss

■ Conditional risk is the expected loss if we take action for a

measurement x.

■Action: = Deciding (i=12)

■Loss:

■Conditional Risks:

■The Overall Risk:

2

1

:i i j i j j

j

R x Ex P x

i

i i

:ij i j

1 11 1 12 2

2 21 1 22 2

R x P x P x

R x P x P x

*

minimization

(minmum value R : Bayes Risk )

R R x x p x dx

(7)

(8)

(9)

(10)

15

Minimum Risk Decision Rule (1)

1 21

2 1 2

if <

if >

R x R x

R x R x

Decide

1 2

21 11 1 12 22 2

Here , <

>

R x R x

P x P x

Minimum Risk Decision Rule (2)

1

1 12 22 2

21 11 12

2

if

Otherwise decide

threshold

P x P

PP x

Decide

(11)

(12)

(13)

16

Fig. 4 Likelihood ratio

17

Minimum error probability decision

=Minimizing the risk with zero-one loss function

Zero-One Loss Function:

1 2

12

Likekihood ratio decision rule (13) becomes

minimum error decisionP x P

PP x

Zero-One Loss Function:

0 if 0 1

, 1 if 1 0

i j ij

i j

i j

(14)

(15)

18

General Framework:

■ Finite set of states of nature (c Classes) :

■ Actions :

■ Loss:

■ Measurement:

1 2, , c

Generalization

: d-dimensional vector (feature vector)x

1 2, , a

: 1,..., 1,...,ij i j i a j c

19

3. Discriminant Function

19

Classifiers represented by discriminant functions : gi(x) i=1,…c

max gi(x)

g1(x) g2(x) gc(x)

x2

…

where arg max

i

jj

i g

x

Classifier minimizing the conditional risk: = i ig x R x

Minimizing error probability: =

Alternate function: =ln ln

i i i i

i i i

g x P x p x P

g x p x P

xd x1 … input

discriminant fnctions

Classifier Network structure

action

20 20

■ Single discriminant function:

Two-category case

1

2

1 2

if 0

if 0

gives the decision boundary

g x

g x

g x g x

Decide

4．Gaussian Case:

1

Multivariate Gaussian: ,

=ln ln

1 1ln 2 ln ln

2 2 2

i i i

i i i

T

i i i i i

p

g x p x P

dx x P

x

(17 )

(18)

(16) 1 2=g x g x g x

21

1

1 1 1

1 1 = ln 2 ln ln

2 2 2

1 1 1ln ln

2 2 2

T

i i i i i i

T T T

i i i i i i i i

dg x x x P

x x x P

0 = T

i i i ig x x Tx W x

1

0

1 1 ln ln

2 2

T

i i i i i iP

Case (i=1,2)

Boundary is given by a linear line

i 1 2General Case

Boundary is quadratic curves

decision boundary

decision boundary

(19)

(20)

1 11where ,

2i i i i i W

22

References: 1) R.O. Duda, P.E. Hart, and D. G. Stork, “Pattern Classification”, John Wiley & Sons, 2nd edition, 2004 2) C. M. Bishop, “Pattern Recognition and Machine Learning”, Springer, 2006 3) E. Alpaydin, Introduction to Machine Learning, MIT Press, 2009 4) A. Huvarinen et. al., ”Independent Component Analysis” Wiley-Interscience 2001

Another action : Rejection

No classification for lower degree of conviction case

What next ? In the discussions so far all of the relevant probabilities are known, but this assumption will not be assured. Fukunaga’s definition of Pattern Recognition: “A problem of estimating density functions in a high–dimensional space and dividing the space into the regions of categories or classes”

23

1

/2 1/2

1

1 1 1, exp

22

is d-dimensional random vector

:

: :

: Determinant of

T

d

T

d

x x

x x x

E x

Cov x E x x

Appendix: Multivariable Gaussian Density Distribution