! "# $% –
(
(a
+
P( x |
i ) dx =
1
b
+
dx
x ai 2
1+ (
)
b
x ai
dx
, du =
b
b
+
dx
1
1
P( x | i )dx =
= (tan 1 u ) + = 1
2
1+ u
u=
+
(b
1
b
1
1+ (
x a1 2
)
b
=
1
b
1
1+ (
x a2 2
)
b
(
x a1 2
x a2 2
) =(
)
b
b
x=
a1 + a 2
2
(c
P(
1
| x) =
P ( x | 1 ) P( 1 )
P ( x | 1 ) P ( 1 ) + P ( x | 2 ) P(
2
)
" x
.
likelihood
!
+, /0
0.5
.
. ( # )*
+,
# )* -
#%
(d
&'
#$%
(1
(a
P(error ) = P (
2
| x) P( x)dx + P(
R1
1
| x) P( x)dx
R2
a1 > a 2
P(error ) =
P(error ) =
a1 + a2
2
0.5
0.5
b
+
0.5
dx
dx
+
x a1 2
x a1 2 a1 + a2 b
1+ (
)
1+ (
)
2
b
b
1
(tan u |
a1 + a2
2
+ tan 1 u | +a1 + a2 ) =
2
1
2
1
tan 1 |
a1
a2
|
2b
(b
" # x 5# 6
7 +,
.
4 0.5
#*
#34 /
4 0.5
" .
-
0.5 # +,
#34 - # a1 = a2 )2 ( c
. ( # )* / 9 .8
(:
(a
P(error ) = P (
2
| x) P( x)dx + P(
P(error ) =
| x) P ( x)dx
1
R2
R1
P( x |
1
) P(
1
)dx +
+
P( x |
2
) P(
2
)dx
(b
P(error ) =
P( x |
1
) P(
1
)dx
P( x |
2
) P(
2
)dx
+
P (error )
P( |
Gaussian @ #6
1
= P( |
) P(
1
1
,A . # )
-
-1
= 0 <#G) H" '
1
) P( |
) = P( |
2
) P(
2
2
) P(
DE
< =
" - # 1 C#"
)=0
; / 9- 4(c
#>
- "
B # # Gaussian @ #6
2
)
? < = %9
. 0
1 C#"
) P(
6 #&) C # "
P( 1|x)
1
/# ;
B#
B F " " ) #* ( d
B # # Gaussian @ #6
8 P(
. %
2|x)
% . " #34
(II
: 8 " J %% =
a
R=
R(
i =1 Ri
O .
i
| x) P ( x)dx
R" N -
4 L #M6 " N
<#G)
L #M6 H" '
:8%
R= [
a
? (a
"/
R(
i =1
i
| x) P (
i
Ri *
( " /0 P# "
/ 9
" N
| x)]P ( x)dx
:8" (b
R ( x) =
a
i =1
P(
i
| x) R(
i
| x)
- # I T5 j
S <#G)
# )* R H" ' /0 " - #
5 j
/0 <#G)
# )*
5 j
S <#G)
S
# )*
S
5 j
" # O . #
/0 " - %; <#G) "
"
)
"
8 % &' "
. #
"-
" E )R" /0
"
S
"
# V#%)
) -J
<#G)
S # )* /
W#
# )*
R.
U#
B
% Q L
= k/ 9? B
-"
<#G)
i=j D E
)
S
&' T
"/
S
-8 % L#P
(c
#2 % Q L .
; -
# S # )* / " 0
#( # H" '
.
)
(IX
R(
R(
i
c
| x) =
s
j =1, j i
| x) =
c +1
( x) =
P(
j
| x) =
s
(1 P (
i
| x)),
i
c +1
r
R(
i
i
| x) = min{R(
j
j
s
(1 P(
i
| x))
c
s
(1 P(
i
| x))
r
(1 P(
P(
| x)}
j
| x))
P(
i
| x) 1
i
| x)
P(
j
| x)
r
s
&'
reject J"
" H" ' /0 " / 9 - %
" # S E 5
/ 9
G reject J#BY H" M 0 "
8
.
) #* % $
4 reject
8
#*
7 ? )
#*
r/ s>1
r=0
R
.
" 8 % <#G)
(Z
1
( x µ1 ) t
2
1
( x µ 2 ) t 1 ( x µ 2 ) + ln P(
2
1 t 1
1
1
1
x
µ1 + µ1t 1 x + ln P( 1 ) = x t 1 µ 2 + µ 2t 1 x + ln P( 2 )
2
2
2
2
t
t
1
1
µ1 x + ln P( 1 ) = µ 2 x + ln P( 2 )
( µ1
µ2 )t
1
1
( x µ1 ) + ln P(
x + ln
P(
P(
)
=0
2)
1
1
)=
R
2
)
# )L#
x ?#[L " \&'
F " ]#&6 ;% ? .
:8" O .
P( 1 )
P( 2 )
P( 1 )
1
µ1 + ln
P( 2 )
( µ1 µ 2 )t
1
( µ1 µ 2 )t
P( 1 )
P( 2 )
P( 1 )
P( 2 )
µ1 + ln
B " 2 \&'
1
0, ( µ1 µ 2 )t
µ1 ) t
1
min{exp{( µ 2
µ1 )t
1
. )L
^ _
µ 2 + ln
µ1}, exp{( µ 2 µ1 ) t
1
µ1}, exp{( µ 2 µ1 )t
1
]#&6
" #
P( 1 )
P( 2 )
P( 1 )
1
µ 2 + ln
P( 2 )
0, ( µ1 µ 2 )t
max{exp{(µ 2
#> - 4
-
J"0
#>
B#
0
0
µ 2 }}
µ 2 }}
] L!
E 5
(`a
(a
ij
=
µ i )( x j
= E[( xi
+ +
+ +
µ i )( x j
( xi
µ j ) P ( xi , x j )dxi dx j
+
µ i )( x j
( xi
µ j )] =
+
µ j ) P ( xi ) P( x j )dxi dx j = ( xi
µ i ) P( xi )dxi ( x j
: % # Gaussian @ E 6 ? " Joint H" '
P ( xi , x j ) =
=
!
2
i
P ( xi , x j ) =
8
1
2
|
=
!
2
i
exp{
2
i
0
2
i
0
1
exp{
i
1/ 2
2 |
ij
2
i
ij
=
1
exp{
j
1 xi
(
2
µi
i
" 2 H" M 0 " . # &'
1
(xi
2
1
,
)2}
=
!1 /
1
1/
1 xj
(
2
exp{
j
B# # #
1
' xi
%
%x
& j
F " " ) #* ( b
µi $
"}
µ j "#
0
0
i
2
2
i
)2
8 R
µj)
µi , x j
µi
1 xi
(
2
b)
µ j ) P( x j )dx j = 0
2
i
µj
j
1 xj
(
2
@E 6#
)2}
µj
) 2 } = P ( xi ) P ( x j )
j
L #M6 b)
X % Q L(c
: 8 " .Y=X2
E[Y ] = E[ X 2 ] =
ij
= E[( X
E[ X 3 ] =
2
0)( X 2
2
+
x 3 N (0,
2
)] = E[ X 3 ]
)dx = 0
.
2
=0
ij
B)
E[ X ] = E[ X 3 ]
E @ #6 /
L -] L
B) /
&'
( cZ
pij - #
.
5 j +, "
" 5i &
;%
# )* ( a
(b
P(
j
| x1 , x 2 ,L , x d ) + P (
ln P (
j
= ln P(
x*
j
ln P(
d
xi
p ij )1
j )( p ij (1
i =1
| x1 , x 2 ,L, x d ) = ln P(
j
)+
d
i =1
)k
j
xi ln
pij
1 pij
j : P(
j
g j ( x) = ln P(
j
j
d
+
i =1
)+
d
i =1
ln(1 pij )
ln P(
k
)*+
P(
k
| x1 , x 2 ,L , x d )
| x1 , x 2 , L, x d )
| x1 , x 2 ,L , xd ) = ln P(
& '(
xi ln pij + (1 xi ) ln(1 p ij )
| x1 , x 2 ,L , x d )
| x1 , x 2 ,L, xd )
xi
j
)+
d
i =1
xi ln
pij
1 pij
, - . / 01
+
d
i =1
ln(1 pij )