PRML BNN まとめ

2. 3 5 71.
2
2
3 5 71.

3. •
•

4. !(#|%, ', () = + )!(#|,, ()!(,|%, ', ( -,
).(#|/(0, ,), (12
).(,|34, 54
6,78(0
.(#| 634
78(0 , (12 + 8 0 7 )548(0 )
12 / 0, , 7 (5, .
6
1 2 4 )3
/ 0, , = :(,7
8(0))
4 )3
, .
6
)!(0) = .(0|;, <12
)!(=|0) = .(=|>0 + ?, @12
6!(=) = .(=|>; + ?, @12 + ><12>7 1

5. 1
!(#, %) '

6. 3 5 71.
2
2
3 5 71.

7. !
"
)$("|!, (, )) = +("|,(!, (), )-.
2(
)$((|/) = +((|0, /-.1
$(2|(, )) = 3
45.
6
)+("4|,(74, (), )-.
)$((|2, /, )) ∝ $((|/)$(2|(, )
. .
,(!, ()1(
3
( .
.
.

8. .
54 :
ln#(%|') = −
+
2
%-
% −
.
2
/
012
3
4(56, %) − 80
9
+ ;<=>8.
%@AB
:
C = −DDln#(%|', +, .) = +E + .F
54 .

9. )"($|&) = (($|$)*+, -./
0(1|2, &) = 30(1|2, $)"($|&)4$
2 ,
5 6, $ ,$ , 4
5 6, $ 7)*+ ) 3 1
( , .
)0(1|6, $, 8) = ((1|5(6, $), 8./
)0(6) = ((6|9, :./
)0(;|6) = ((;|-6 + =, >./
?0(;) = ((;|-9 + =, >./ + -:./-@
2
5

10. !"($, &) ≃ "($, &)*+) + -.(& − &)*+
- = 1&"($, &)|&3&456
7(8|9, :) = ;7(8|9, &)<(&|:)=&
7(8|9, >, ?) ≃ @(t|" $, &)*+ + -. & − &)*+ , ?BC)
)<(&|:) = @(&|&)*+, DBC
" $, & E)*+
,

11. )"($) = &($|(, *+,
)"(-|$) = &(-|.$ + 0, 1+,
2"(-) = &(-|.( + 0, 1+,
+ .*+,
.3. 3
"(4|5, 6, 7) ≃ &(t|: $, ;<=> + ?3
; − ;<=> , 7+,
)
)A(;|B) = &(;|;<=>, .+,
)"(4|$, B) = &(4|: $, ;<=> , CD($)
CD($) 7+, ?3.+,?
. 3
=4 8 + ; =:(5E, ;)
3

12. 3 5 71.
2
2
3 5 71.

13. B !, #. )
$(&|!, #) = *$(&|+, #)$(+|!),+
ln$(&|!, #) ≃ −1(+234) −
1
2
ln|7| +
9
2
ln! +
:
2
ln# −
:
2
ln(2;)
1(+234) =
#
2
<
=>?
@
A(B=, +234) − C=
D
+
!
2
+234
E
+234
2
0 1 2
(2 )
( ) ( )
W
Z f z dz f z
A
p
= =ò
F(+) $(&|+, #)$(+|!)
G $(&|!, #)
F(+234) $(&|+234, #)$(+234|!)
C
314 (
314 15 ) =

14. !, #
#$%& = (%&
$ )*+,.
! =
-
)*+,
. )*+,
1
W
i
i i
l
g
l a=
=
+
å
ln1(3|!, #) ≃ −
#
2
9
:;<
=
>(?:, )*+,) − @:
A
−
!
2
)*+,
.
)*+, −
1
2
ln|C| +
E
2
ln! +
F
2
ln# −
F
2
ln(2G)
!, #.
ln|C|2
C = !H + #$
3
1
#
=
1
F − -
9
:;<
=
>(?:, )*+,) − @:
A
• !, #2
5 3
!, #2 2 3

15. • !, #
!, #
)%('|)
'*+, '
)-('|'*+,, ./0
'*+, %()|!, #)
12%()|!, #) !, #
!, # 3!, 4#
3!, 4#

16. 5
5 !(#| %&, ())
15 .
1 1
! # %&, () ∗ 3! 2/ ! # %&, () ∗ 4! 21
. 2345
6! 23
5 1

17. 3 5 71.
2
2
3 5 71.

18. .
ln#(%|') = *
+,-
.
/+ln0+ + (1 − /+)ln(1 − 0+)
}/+ ∈ {0,1
)0+ = 0(9+, '
0
:;
:-
:<
. =
)#('|>) = ?('|@, >A-B
C-
CD

19. 8 4 !"#$ .
5 % )
.
•
)'(!|!"#$, +,-
• 8 4.
ln1(2|.) ≃ −5(!"#$) −
1
2
ln|8| +
:
2
ln.
5(!"#$) = <
=>-
?
@=lnA= + (1 − @=)ln(1 − A=) +
.
2
!"#$
B
!"#$
)A= = A(C=, !"#$
. =
D
!"#$
B
!"#$
. (

20. !(#|%, ') = *!(#|%, +),(+|')-+
! # ., + = / %0+ = 1(2 %, + )
5 ,
32(%, +) ≃ 2(%, +567) + 9:(+ − +567
9 = <2(%, +567)
!(2|%, ') = *=(2 − 2 %, +567 − 9:
+ − +567 ),(+|')-+
!(2|%, ') = >(2|2 %, +567 , 1?
@
(%))
1?
@(%) = 9:(%)ABC9(%)
!(# = 1|%, ') = *1(2)!(2|%, ')-2
! # = 1 %, ' = )1(E(1?
@)2567
E .(4.154)
42
4

21. ! " = 1 %, ' = ))(+(),
-)./01! " %, 2/01 = )(. %, 2/01 )
. , 5 9 A
01
,