Machine leanring chapter no 3

1. INTRODUCTION
TO
MACHINE
LEARNING
3RD EDITION
ETHEM ALPAYDIN
© The MIT Press, 2014
alpaydin@boun.edu.tr
http://www.cmpe.boun.edu.tr/~ethem/i2ml3e
Lecture Slides for

2. CHAPTER 3:
BAYESIAN DECISION
THEORY

3. Probability and Inference
3
 Result of tossing a coin is {Heads,Tails}
 Random var X {1,0}
Bernoulli: P {X=1} = po
X (1 ‒ po)(1 ‒ X)
 Sample: X = {xt }N
t =1
Estimation: po = # {Heads}/#{Tosses} = ∑t
xt / N
 Prediction of next toss:
Heads if po > ½, Tails otherwise

4. Classification
 Credit scoring: Inputs are income and
savings.
Output is low-risk vs high-risk
 Input: x = [x1,x2]T ,Output: C Î {0,1}
 Prediction:















otherwise
0
)
|
(
)
|
(
if
1
choose
or
otherwise
0
)
|
(
if
1
choose
C
C
C
C
,x
x
C
P
,x
x
C
P
.
,x
x
C
P
2
1
2
1
2
1
0
1
5
0
1
4

5. Bayes’ Rule
 
   
 
x
x
x
p
p
P
P
C
C
C
|
| 
   
         
    1
1
0
0
0
1
1
1
1
0














x
x
x
x
x
|
|
|
|
C
C
C
C
C
C
C
C
P
p
P
p
P
p
p
P
P
5
posterior
likelihood
prior
evidence

6. Bayes’ Rule: K>2 Classes
     
 
   
   




K
k
k
k
i
i
i
i
i
C
P
C
p
C
P
C
p
p
C
P
C
p
C
P
1
|
|
|
|
x
x
x
x
x
   
   
x
x |
max
|
if
choose
and
1
k
k
i
i
K
i
i
i
C
P
C
P
C
C
P
C
P


 

1
0
6

7. Losses and Risks
 Actions: αi
 Loss of αi when the state is Ck : λik
 Expected risk (Duda and Hart, 1973)
   
   
x
x
x
x
|
min
|
if
choose
|
|
k
k
i
i
k
K
k
ik
i
R
R
C
P
R






 
1
7

8. Losses and Risks: 0/1 Loss






k
i
k
i
ik
if
if
1
0

   
 
 
x
x
x
x
|
|
|
|
i
i
k
k
K
k
k
ik
i
C
P
C
P
C
P
R








1
1


8
For minimum risk, choose the most probable class

9. Losses and Risks: Reject
1
0
1
1
0










 


otherwise
if
if
,
K
i
k
i
ik
   
     
x
x
x
x
x
|
|
|
|
|
i
i
k
k
i
K
k
k
K
C
P
C
P
R
C
P
R










1
1
1




     
otherwise
reject
|
and
|
|
if
choose 




 1
x
x
x i
k
i
i C
P
i
k
C
P
C
P
C
9

10. Different Losses and Reject
10
Equal losses
Unequal losses
With reject

11. Discriminant Functions
  K
i
gi ,
,
, 
1

x
   
x
x k
k
i
i g
g
C max
if
choose 
   
 
x
x
x k
k
i
i g
g max
| 

R
 
 
 
   






i
i
i
i
i
C
P
C
p
C
P
R
g
|
|
|
x
x
x
x

11
K decision regions R1,...,RK

12. K=2 Classes
 Dichotomizer (K=2) vs Polychotomizer (K>2)
 g(x) = g1(x) – g2(x)
 Log odds:
 


 
otherwise
if
choose
2
1 0
C
g
C x
 
 
x
x
|
|
log
2
1
C
P
C
P
12

13. Utility Theory
 Prob of state k given exidence x: P (Sk|x)
 Utility of αi when state is k: Uik
 Expected utility:
   
   
x
x
x
x
|
max
|
if
Choose
|
|
j
j
i
i
k
k
ik
i
EU
EU
α
S
P
U
EU




 
13

14. Association Rules
 Association rule: X  Y
 People who buy/click/visit/enjoy X are also
likely to buy/click/visit/enjoy Y.
 A rule implies association, not necessarily
causation.
14

15. Association measures
15
 Support (X  Y):
 Confidence (X  Y):
 Lift (X  Y):
 
 
 
customers
and
bought
who
customers
#
#
,
Y
X
Y
X
P 
 
 
 
 
X
Y
X
X
P
Y
X
P
X
Y
P
bought
who
customers
and
bought
who
customers
|
#
#
)
(
,


 
)
(
)
|
(
)
(
)
(
,
Y
P
X
Y
P
Y
P
X
P
Y
X
P



16. Example
16

17. Apriori algorithm (Agrawal et al.,
1996)
17
 For (X,Y,Z), a 3-item set, to be frequent (have
enough support), (X,Y), (X,Z), and (Y,Z) should
be frequent.
 If (X,Y) is not frequent, none of its supersets
can be frequent.
 Once we find the frequent k-item sets, we
convert them to rules: X, Y  Z, ...
and X  Y, Z, ...