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Homework solution reference for CPE646
Typology: Assignments
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Homework 2 solutions
Problem 1.
P1.1 Maximum likelihood estimation
2
otherwise
x
xe x
p x
2
1 1
1 1 1 2 1 2
2
1
2
1
( ) ln ( | ) ln ( | ) ln ( | )
ln ( | )
ln
ln
we have
f
ln
or and
k
n n
k k
k k
n k k n k n k n k n k n k
x
k
k k
x e
x
l p D p x p x
l p x
x
n
x
x x
n
x
P1.2 Bayesian estimation. Given
We estimate
( ) ~ ( , ) , 0 and fixed
otherwise
p U
2
1
1 1
let ( | ) ( ) = , which is a normalization factor independent of
we try to find whi
k
x
n
k
k
n n
k
k k
k
p x p
p D p
p D
p D p d p D p d
p D p d
p D p x p x e
1 1 2 1 2 1
2
2
2
ch maximizes ( | ) as our estimate of
ln ( | ) ln ln ln ln
ln
ln ( | )
ln ln
we have
for and 0<
because ln ln ln isu
k k
k k
k
n k n k n k n k k
x x
x x
p D
p D
p
x
x
x x
n
x
nimodal and increases before maxima
so if > we let =
We can see that if we use y k
for classification, which is the result of LDA for dimension
reduction, the classification performance will be good.
1
2
1 2
1 1
m = ,m =
i
i
W
t
i i
x D
x m
x m
1
1
1 2
within-class scatter matrix,
W
t
k k
w S m m
y w x
the reconstructed da
i
t s
a
k k
x y w
