[prml] パターン認識と機械学習（第2章：確率分布）

2.1

2.1.1 2.2

2.1.1 2.3

2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6

1

2.3

2.3.7 t 2.3.8 2.3.9

2.4 2.4.1 2.4.2 2.4.3

2.5 2.5.1 2.5.2

2

2.1 x 0, 1 1 x

Bern(x) = (1 )

=0.7 http://qiita.com/katsu1110/items/b0213c7ef6a8122abfc5

{ }

x 1x

3

http://qiita.com/katsu1110/items/b0213c7ef6a8122abfc5

E[x] =

var[x] = (1 )

4

2.1 N x = 1 m

Bin(mN ,) = (1 )

https://ja.wikipedia.org/wiki/

(Nm

) m Nm

5

https://ja.wikipedia.org/wiki/%E4%BA%8C%E9%A0%85%E5%88%86%E5%B8%83

E[x] = N

var[x] = N(1 )

6

2.1.1

Beta(a, b) = (1 )


(a)(b)(a+ b) a1 b1

7

https://ja.wikipedia.org/wiki/%E3%83%99%E3%83%BC%E3%82%BF%E5%88%86%E5%B8%83

E[] = , var[] =

abx = 1x = 0

a+ ba

(a+ b) (a+ b+ 1)2ab

8

2.2

9

N

Mult(m ,m , ,m ,N) =

K = 2

1 2 K (m m m1 2 K

N )k=1

K

kmk

10

2.2.1

Dir(,) =

3 = 0.1, = 1, = 10

( )( )1 K

( )0

k=1

K

k 1k

{ k} { k} { k}11

2.3 Dx

N (x,) = exp (x ) (x )

D

D D

(2)D/21

1/21 {

21 T 1 }

12

= (x ) (x )

{ T 1 }1/2

13

D1

E[x] =

2

E[xx ] = +

E[xx ] E[x] =

T T

T 2

14

D(D + 3)/2 D(D + 1)/2

D

1

15

WoodburyABCD

(A+BD C) = A A B(D + CA B) CA

Woodbury3

A C B

1 1 1 1 1 1 1

16

2.3.1

p(x x ) = N (x , )

= + (x )

=

a b a ab aa1

ab a ab bb1

b b

ab aa ab bb1

ba

17

A,B,C,D

=

Schur complement matrixM

M = (ABD C)

(AC

BD

)1

( MD CM1

MBD1

D +D CMBD1 1 1)

1 1

18

2.3.2

p(x ) = N (x , )

E[x ] =

cov[x ] =

a a a aa

a a

a aa

19

2.3.3 xxy

p(x) = N (x, )

p(yx) = N (yAx+ b,L )

p(y) = N (yA+ b,A A )

yyx

p(xy) = N (x A L(y b) + ,)

= (+A LA)

1

1

1

{ }

1

20

2.3.4

ln p(x,) = ln (2) ln (x ) (x )

0

= x

0

= (x )(x )

2ND

2N

21

n=1

N

n 1

n

ML N

1

n=1

N

n

MLN

1

n=1

N

n ML n ML

21

2.3.5

= + (x )ML(N)

ML(N1)

N

1N ML

(N1)

22

2.3.6

23

1

p(x) = p(x ) = exp (x )

p() = N ( , )

2

n=1

N

n(2 )2 2

N

1 {221

n=1

N

n2}

0 02

24

1

p(x) = p(x ) = exp (x )

Gam(a, b) = b exp (b)

n=1

N

n1 2

N {2

n=1

N

n2}

(a)1 a a1

25

1

p(x,) = exp (x )

exp exp x x

p(,) exp exp (c d)

= exp ( ) exp d

n=1

N

(2 )

21

{2

n2}

[ 21 (2

2 )]N

{n=1

N

n 2

n=1

N

n2}

[ 21 (2

2 )]

{2

c 2} 2

{ (2c2 ) }

26

-a = (1 + )/2b = d c /2

p(,) = N ( , () )Gam(a, b)

2

01

27

DD N (x, )

1

28

1 D

29

2.3.7 t 1

St(x,, ) = 1 +

= 1

(/2)(/2 + 1/2) (

)1/2

[

(x )2 ]2

21

30

t t

31

2.3.8

2x

32

mI (m)

p( ,m) = exp m cos( )


0 0

0 2I (m)0

1{ 0 }

33

https://ja.wikipedia.org/wiki/%E3%83%95%E3%82%A9%E3%83%B3%E3%83%BB%E3%83%9F%E3%83%BC%E3%82%BC%E3%82%B9%E5%88%86%E5%B8%83

2.3.9

2

34

p(k) = N (x , )

= 1, 0 1

k=1

K

k k k

k=1

K

k k

35

2.4

p(x) = h(x)g() exp u(x)

g() h(x) exp u(x) dx = 1

{ }

{ }

36

Bern(x) = (1 ) = (1 ) exp ln

() =

u(x) = xh(x) = 1g() = ()

p(x) = () exp (x)

x 1x { (1

)}

1 + exp ()1

37

Mult(x) = = exp x ln

u(x) = xh(x) = 1g() = 1

p(x) = exp ( x)

k=1

M

kxk {

k=1

M

k k}

38

2.4.1

p(x) = h(x)g() exp u(x)

u(x) n g n

http://s0sem0y.hatenablog.com/entry/2016/05/25/025947

{ }

39

http://s0sem0y.hatenablog.com/entry/2016/05/25/025947

2.4.2 f(, )

p(, ) = f(, )g() exp { }

40

2.4.3

41

A BA c B c

p(x) = f(x )

p( c) = p()p()

42

A BA/c B/c

p() = p

p() 1/0

(c

1 )

1

43

2.5 ex.

44

3DMD

M

45

D p(x) N p(x)

R P K R

K NP

p(x) R

P p(x)V

46

_2RVK

p(x) =

KVVKK

NV

K

47

2.5.1

48

2.5.2 h

p(xC ) = K /N Vp(x) = K/NVp(C ) = N /N

p(C x) = =

k k k k k

kp(x)

p(xC )p(C )k kK

Kk

49

[prml] パターン認識と機械学習（第2章：確率分布）

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