Download Homework 1 Solutions: Probability and Decision Theory and more Assignments Pattern Classification and Recognition in PDF only on Docsity!
Homework 1 Solutions
%P1.
x=-10:0.01:10;
y1=gauss(0,1,x');
y2=gauss(2,4,x');
plot(x,y1,x,y2);
print -djpeg hw1_1_1.jpg
y1_5=gauss(0,1,5)
y1_5 =
1.4867e-
y2_5=gauss(2,4,5)
y2_5 =
( , ) ~ ( ) exp
P1.1 Likelihood
( | ) exp
( | ) exp
x
N p x
p x x
x
p x
P1.2 Prior
Evidence
P P
p x p x P p x P
%P1.
e=0.6y1+0.4y2;
plot(x,e)
print -djpeg hw1_1_2.jpg
e_5=0.6gauss(0,1,5)+0.4gauss(2,4,5)
e_5 =
P1.3 Posterior
P x p x P p x
P x p x P p x
%P1.
l=y1./y2;
plot(x,l)
print -djpeg hw1_1_4.jpg
l_5=gauss(0,1,5)/gauss(2,4,5)
l_5 =
2.2958e-
12 2 2
P1.6 Likelihood ratio threshold
For defined los
s fun
ti n
c o
P P
P P
1 12 22 2
2 21 11 1
11 1 1 12 1 2
21 2 1 22 2 2
1
For general loss function, decide if:
P1.5 Likelihood r
atio threshold
For zero-one loss
f
p x P
p x P
a a
a a
2
1
unction
P
P
%P1.
th=4/3;
c1=(l>=th);
c2=(l<th);
plot(x,c1,x,c2)
print -djpeg hw1_1_6.jpg
1 1 1 11 1 12 2 2
2 2 2 21 1 22 2 1
1 2
In region : ( | ) ( | ) ( | ) ( | )
In region : ( | ) ( | ) ( | ) ( | )
The over all risk is
P1.7 Bayes
risk
R x P x P x P x
R x P x P x P x
R R x p x dx R x p x dx
1 1
1 2 1 2
1
1 2
1
( )
t -
2 1 1 2 2 1
1 2
1 2
if ( | ) are normal
( | ) ( | ) , where
( ) ( - ) [ +(1- ) ] ( - )
+ ln
In Bh
P2.2.
attac
i
k
P error P P p x p x dx
p x
p x p x dx e
k
1
1 2
1 2
0
t -1 1 2
2 1 1 2 2 1
1 2
( / )
haryya bound, /
( / ) / ( - ) [( + )/2] ( - )+ ln
*. [ , ] ln
k
k
P error P P e
e
6779
.
