PRML エビデンス近似

2. 3.5
3.5.1
3.5.2
3.5.3
3.6

3. 3.5
3.5.1
3.5.2
3.5.3
3.6

4. )"($, &) = &(
)($

5. !(#|%, ', () = + )!(#|,, ()!(,|%, ', ( -,
)!(#|., %, ', () = /(#|01
2
3(.), 41
5
(.)
41
5
(.) =
6
7
+ 3 . 2913(.) 91
:6
= '; + (<=<
', (
>1 = (91<=%

6. !(#|%, ', () = + )!(#|,, ()!(,|%, ', ( -,
!(#|%) = .!(#|,, ()!(,|%, ', ()!(', (|%) -,-'-(
' (
', ( !(', ()
,

8. !(#|%) ≃ !(#|%, )*, +,) = . /!(#|0, +, !(0|%, )*, +,)10
*, , )*, +, ()
!(#|%) = 2!(#|0, ,)!(0|%, *, ,)!(*, ,|%) 101*1,
*
,
!(*, ,|%)
)*
+,
*, , )*, +,
A

10. 3.5
!(#, %|') ,
!()|') = +!()|,, %)!(,|', #, %)!(#, %|') -,-#-%
)!(#, %|') ∝ !('|#, %)!(#, %
!('|#, %), /#, 0%,
, ,
, .
!('|#, %),

11. .
!(#|%, ') )%, *'
+
.
,

12. ! " #, %
• ,
: , #, %
•
)!(#, %|") ∝ !("|#, %)!(#, %
!("|#, %) = +!("|,, %)!(,|#)-, !(,|.) =
)!(.|,)!(,
)!(.

13. 3.5
3.5.1
3.5.2
3.5.3
3.6
. .

14. 3.5.1
, ", # ,
,
,
ln&((|", #) =
,
-
ln" +
/
-
ln# − 1(2/) −
3
-
ln|4| −
/
-
ln(26)

15. 5 2 ). (
2P
5 9 2 ).
!(#|%, ') = *!(#|+, ')!(+|%),+
-
./0
1
)2(34|+56(74), '80 )2(+|9, %80: + ,10

16. !(#|%, ') =
'
2+
⁄- .
%
2+
⁄/ .
0 }exp{−7(8) 98
)7(8) = '7:(8) + %7<(8
=
'
2
‖# − >8‖. +
%
2
8?8
!(#|%, ') = 0!(#|8, ')!(8|%)98
@
A
B
7:(8)=
7<(8)=
C
.
8D
8
1
2
F
GHC
-
IG − 8JK(LG) .

17. = "($%) +
(
)
* − $%
,-(* − $%)
- = ./ + 012
1
0
2
‖5 − 6$%‖) +
.
2
$%
,$%
)- = 77"(*
$% = 0-89
1:5
;(5|., 0) = > }exp{−"(*) D*
( ) P
"(*) =
0
2
‖5 − 1*‖) +
.
2
*,*
21
$%, - 5 );(*|5, ., 0
$%, E%
8(
;(*|F) =
);(F|*);(*
);(F

18. ! }exp{−((*) ,*
= exp −( ./ ! exp −
1
2
* − ./
2
3(* − ./) ,*
= exp −( ./ 24 ⁄6 7|9| ⁄:; 7
((*) = ((./) +
;
7
* − ./
23(* − ./)

19. ln#(%|', )) =
,
-
ln' +
/
-
ln) − 1(2/) −
3
-
ln|4| −
/
-
ln(26)
#(%|', )) =
)
26
⁄/ -
'
26
⁄, -
8 }exp{−1(>) ?>
exp −1 2/ 26 ⁄, -|@| ⁄A3 -

21. 3.5
3.5.1
3.5.2
3.5.3
3.6

22. ln#(%|', )) =
,
-
ln' +
/
-
ln) − 1(2/) −
3
-
ln|4| −
/
-
ln(26)
4 = '7 + )898
2/ = )4:;
8<%
)8=8 >? = @?>?
1
9
9 3 9
1 0
0 M
l a
l a
+æ ö
ç ÷
ç ÷
ç ÷+è ø
!
" # "
$
1 2/ =
)
2
‖% − B2/‖-
+
'
2
2/
=
2/
' + C
?D3
/
)EF(G?)EF(GH … C
?D3
/
)EF(G?)E,:3(G?
⋮ ⋱ ⋮
C
?D3
/
)E,:3(GH)EF(GH ⋯ ' + C
?D3
/
)E,:3(G?)E,:3(G?
(M, M)

23. !
"
"!
ln % =
"
"!
ln '
()*
+
,( + ! =
"
"!
.
()*
+
ln ,( + !
1
i il a
=
+
å
0 =
0
2!
−
1
2
45
6
45 −
1
2
7
(
1
,( + !
ln8(:|!, =) =
+
?
ln! +
5
?
ln= − @(45) −
*
?
ln|A| −
5
?
ln(2B)
@ 45 =
=
2
‖: − D45‖?
+
!
2
45
E
45
!45
6
45 = 0 − 7
(
!
,( + !
= F F = 7
(
,(
! + ,(
! =
F
45
G
45

24. !
! =
#
$%
& $% ' = !( + *+,+
$% = *'-.
+/0
x
1 f!
2+,
= (4(15), ⋯ , 4(1%)
* 0
*+,+
' = !( + *+,+
$% = *'-.
+/0
! =
#
$%
&
$%
i
i i
l
g
a l
=
+
å
9::+,+
1
:
'

25. !
1
ln ln( ) i
i
i i i
d d
d d
l g
l a
b b b l a b
A = + = =
+
å å
0 =
$
2!
−
1
2
(
)*+
,
).) − /,0(2)
3 −
4
2!
1
!
=
1
$ − 4
(
)*+
,
).) − /,
5
0(2)
3
i id
d
l l
b b
=
ln8(9|;, !) =
=
3
ln; +
,
3
ln! − ?(/,) −
+
3
ln|@| −
,
3
ln(2A)
!BCB DE

26. !
" f#
$%&
= ()("*), ⋯ , )(".)
! /
!%&%
0 = #1 + !%&%
3. = !045
%6/
# =
7
3.
8
3.
i
i i
l
g
a l
=
+
å
9::
1
!
=
1
< − 7
>
?@*
.
)A? − 3.
8
B("?
C
1
!
=
1
< − 7
‖A − %E3.‖C
# =
7
3.
8 3.
# !
# ! ,
,#
!

27. 1
6 2 2
1
!"#(%| '(, *+)

28. 3.5
3.5.1
3.5.2
3.5.3
3.6

29. !(#|%) = (!(#|), +)!()|%, ,, +)!(,, +|%) -)-,-+
)!(,, +|%) ∝ !(%|,, +)!(,, +
!(%|,, +) 3 ̂,, 0+.
!()|%, ,, +) )∝ !(%|1, ), +)!()|, .
,, + 53 1 .
!(%|,, +) = 2!(%|), +)!()|,)-)

30. !" < !$
!"
!$
%&'
((*|,)
((-|., *, 0)
2
34"
5
)6(78|*9:(.8), 0;"
= −
0
2
?
84"
5
78 − *9
:(.8) $@A(*)
0BCB D3 = !3D3
3
1 (
BC
B1 .3
) 3 1 .
= −
0
2
‖- − B*‖$
"
FG
<
"
FH
((*|-, ,, 0) )∝ ((-|., *, 0)((*|,

31. !
! = 0
$(&|(, !, *) )∝ $((|-, &, *)$(&|!
./0
MAPw
1(&) =
*
2
‖( − 5&‖6
+
!
2
&8
&
$(&|(, !, *) ∝ ( )
2 2
exp ( )
2 2
N M
E w
b a
p
æ ö æ ö
-ç ÷ ç ÷
è ø è ø

32. !
0 ≤
$%
$% + !
≤ 1
0 ≤ ( ≤ )
2l a!
1l a!
i
i i
l
g
a l
=
+
å
$*
$+
$*
$* + !
≃ 0
$+
$+ + !
≃ 1
-+ -./ 0
-* 0
0(2|!)
0(4|5, 2, 7)
$* ≪ ! ≪ $+
0a ¹
mode 2>?@ = BC

33. !" !#$
2l a! %(') =
*
2
‖- − /'‖" +
1
2
'2'
!"3K ,
90 2 21
1
*/3/
4 = 15 + */3
/
67 = *489
/:-
1 =
;
67
<
67
i
i i
l
g
a l
=
+
å
=>:
2 2 2MAP MLw wg=
() ( (

34. !"
#(%) =
(
2
‖+ − -%‖. +
0
2
%1%
1l a! !"
!"
)2(%|4, 06"7

35. - !" W
le
il a!
i
i i
l
g
a l
=
+
å W m
W
!" W W n
W r W m #
i M − #
≫
N d W - t M
)( = *+,(.

36. !" !#
– !#, !" %# %"
!"
– %" %&' .
– ))"(+, -,
!#
– ))#(+, -, . .
%#
– %# ,
-,
))"(+,
%"
0 1

37. !
– !
1
!
=
1
$
%
&'(
)
)+& − -)
.
/(1&
2
2 2
1
1
( )
1
N
MAP n ML
n
x
N
s µ
=
= -
-
å
2 2
1
1
( )
N
ML n ML
n
x
N
s µ
=
= -å
3
1
!
=
1
$ − 3
%
&'(
)
)+& − -)
.
/(1&
2

38. ! − #
2 2
1
1
( )
1
N
MAP n ML
n
x
N
s µ
=
= -
-
å
,
,
1
%
=
1
! − #
'
()*
+
)-( − .+
/
0(2(
3
#
, ,
1{ , }Nx x!4*, ⋯ 4+ 7
4*, ⋯ 4+8*
4+ 7 ! − 1
! , k
! − k
,

39. O !" O
#$# r e %"
& = ( ) =
(
)2,-(/0
1 =
2
)2,3(/0
1
1
=
1
2 − &
6
789
0
):7 − /0
;
<(=7
>
) =
&
/0
; /0
),3(/0 =
1
2
‖@ − A/0‖>
),-(/0 =
1
2
/0
B/0
M i e K W
N M!
il a!
O l n e d
m !CDE = !CF -

40. 1.1
!M 10
0
2# )%&(()
*+ ,#
ln/(0|#, 3):
*+ ,#
i
i i
l
g
a l
=
+
å # =
!
()
5 ()
,#

41. !" #
$=0$
i
i i
l
g
a l
=
+
å
$
!" )&((|*, $,-.
!"
$
!" !/0
1(() =
3
2
‖6 − 8(‖9 +
$
2
(;(

42. 3.5
3.5.1
3.5.2
3.5.3
3.6 ….

43. 3
)"($, &) = &(
)($,
2
&1 ( 1 2 )
2
1 $2 .
.3 2

44. 3 2 7
• ! "#$(&
)&( = (*(, ,(), &- = (*-, ,-
#. &( = 0.8
#((&() = 0.3
#- &( = 0.0
#3(&() = 0.5
#4(&() = 0.0
#. &- = 0.0
#((&-) = 0.0
#- &- = 0.0
#3(&-) = 0.0
#4 &- = 0.3
#5 &- = 0.0
#6(&-) = 0.0
#7(&-) = 0.6
#9(&-) = 0.0
*
,
*
*(
,(
*( *-
#. *( = 0.8
#((*() = 0.3
#-(*() = 0.0
#. #( #-
#. *- = 0.0
#((*-) = 0.3
#-(*-) = 0.6
#.
#(
#-
…
*-
,-
#3
#4
#5
#6
#7
#9
"#$(& : 1

45. RB 5 ,
• RB E !" 5 ,
• # 5 1 R E R
$%
$&
$'
• x 5 RB 5 {2.
) E R RE E E
F
• x 2 1 {1
) R E R

46. )"($, &) = &()($
*+, ,-
*+, ,- &