【目次】第二章:確率分布 2.1 確率分布 2.1.1 ベータ分布 2.2 多値変数 2.1.1 ディリクレ分布 2.3 ガウス分布 2.3.1 条件付きガウス分布 2.3.2 周辺ガウス分布 2.3.3 ガウス変数に対するベイズの定理 2.3.4 ガウス分布の最尤推定 2.3.5 逐次推定 2.3.6 条件付きガウス分布 1
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 )
https://ja.wikipedia.org/wiki/
(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( )
https://ja.wikipedia.org/wiki/
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