ベイズロジスティック回帰

2. 1
. . 2
4 5

3. !(#|%) =
)!(%|#)!(#
)!(%
!((|#) = )
*+,
-
.*
/0
1 − .*
,3/0
4.* = !(5,|6*) = 7(896*

4. !(#|%) =
)!(%|#)!(#
)!(%
!(#|%) ∝ )!(%|#)!(#
.

5. 1
4
4 .1
1
1

6. !(#|%) ∝ ! % # !(#)
)((#|)*, ,-.
/01
2
3/
45
1 − 3/
1845
ln!(#|;) = −
1
2
(# − )*),*
81
(# − )*) + ?
/01
2
@/ln3/ + (1 − @/)ln(1 − 3/) + ABCD@

7. MAP $%&'
-
$%&'
(E * = −(ln/(*|2) = 45
67
(* − 85) + :
;<7
=
(>; − ?;)@;

8. !"
#$
= −''ln*(,|.) = !0
#$
+ 2
34$
"
53(1 − 53)7373
8
*(,|9) ≃ );(,) = <(,|,=>?, !"

9. !(#) . %&
'(%&|), +) = -'(%&|), .)'(.|+)/.
≃ - )1(.2) 3(.)/.
' %4 ), + = 1 − '(%&|), +)
)1(.2) ) .

10. ! = #$%
)'(#$% = )*(! − #$% )'(!),!
) )'(#$% -(#),# = . )*(! − #$% )'(!),! -(#),#
= . )*(! − #$
% )-(#),# '(!),!
= )'(!)/(!),! / ! = )*(! − #$% ) -(#),#
#$%

11. ! )#(%)&(%)'% = &(0!
*+
+
#(%)'% = 1
-#(. − %)&(%, 1)'%'1 = !&(., 1)'1
2 . = -#(. − % − 1)&(%, 1)'%'1 = -
3*4*567
&(%, 1)'%'1
.
%
1
1,0
%
1 = .
1,0 1
&(%, 1)
1,1
%
1 = .1,1
.
# % = 0 (% ≠ 0)
= ∞ (% = 0)

12. ! "
! " = $%(" − ()* ) ,(()-(
,(() *
,(() , ! "
! "
./, ⋯ , .2
3/
⋮
32
= "
(

13. ! "
#$ = &$["] = )!(")","
= - ).(" − 01
2 ) 3(0),0 ","
= -3(0) ).(" − 012 )"," ,0
= )3(0)012 ,0
= )3(0)0,0
4
2 = 0567
1
2
) ).(8 − 9):(8),8 = :(9
: 8 = 8
)3(0) = ;(0|0567, >?

14. ! "
#$
% = var$["] = ,$["%] − ,$ " %
= .!(")"% 1" − 2345
6
7
%
= 8 .9(" − 2:7 ) ;(2)12 "%1" − 2345
6
7
%
= 8;(2) .9(" − 2:
7 ) "%
1" 12 − 2345
6
7
%
= .;(2)(2:7)% 12 − 2345
6
7
%
= .;(2)76226712 − 7623452345
6
7
= 76 .;(2) 22612 7 − 7623452345
6
7
= 7:var2[2]7
= 7<=>7
= 76 .;(2) 226
12 −2345 2345
6
7
.;(2) 226
12 = Ε[226
]
Ε[226
] = 23452345
6
+ =>

15. )"($|&', )'
*
+ ,- ., / ≃ 1 ))(23
. 4(2)52 = 1) $ + $ 5$ = 1 ))($)"($|&', )'
*
5$ $ = 23
.
)"($|&', )' 5 ( 30. 8 )
7+(8|9, :, ;, <) = "(8|=>
?
@(9), )>
*
(9) (3.58)
)"($|&', )'
*
= "(2?
.|2EFG
?
., .?
H>.)
)"($|&', )'
* = 1I($ − 23. )4(2)52
+(8|:, ;, <) = 1+(:|2, <)+(2|8, ;, <)52
)"(:|23., <K-
(3.57)

16. Φ
"
8
$
% $
)Φ(($ % $
() =
+
,
- ./ 0, 2 ≃ 4 )%(56
0 7(5)85 = 4% $ - $ 8$ = 4 )%($)9($|;<, %< 8$

17. ! )Φ(%&)'(&|), +, -& = Φ
)
%/, + +, 12
,
)+(&) ≃ Φ(%&
! )+(&)'(&|), + -& ≃ +
%/2)
%/, + +, 12
,
= +
)
1 + %,+,
2
,
= +
)
1 +
5+,
8
2
,
%,
=
5
8
7(+,
) = 1 + ⁄5+,
8 / ⁄2 , = + 7(+,
))
! )+(&)'(&|)9, +9 -& ≃ + 7(+9
,
))9

18. ! "# $, & = 0.5
+(-.
/ )1.
- +(-.
/ )1.
+ -.
/ 1. = 0
+(-/) = 1 + ⁄5-/ 8 7 ⁄# /
1. = 0 ⇔ 9:;<
=
$ = 0
9:;<
! "# $, 9 = - 9:;<
=
$ = 0.5
!("#|$, &) ≃ @ )-(9A
$ B(9)C9

19. !P ,
,
)#(%, '() > #(%, '* % '( A .
)#(%, '+) = #('+|%)#(%
%
#('+|%) A .
. M A
# '( ., / ≃ 1 )2(!3
. 4 ! 5! = 1 )2(6)7(6|89, 29 56 ≃ 2 :(29
*
)89

20. !(#|%)
' ((')
* )+)