Shmeioseis beyzianis statistikis
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Transcript of Shmeioseis beyzianis statistikis
-
, 2004
-
v
1 1
1.1 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 . . . . . . . . . . 4
1.3 . . . . . . . . . . . . . . . . . . . . . 6
1.4 . . . . . . . . . . . . . . . . . . . . . . . 10
1.5 . . . . . . . . . . . . . . . 13
1.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 17
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 . . . . . . . . . . 18
2.2.1 . . . . . . . . . . . . . . . . . 18
2.2.2 . . . . . . . . . . . . . . . . . . . . . . 19
2.2.3 . . . . . . . . . . 21
2.3 . . . . . . . . . . . . . . . . . . . . . . 23
2.4 . . . . . . . . . . . . . . . . 28
2.4.1 ( ) . . . . . . 29
2.4.2 . . . . . . . . . . . . . . . . . 31
2.4.3 Jeffreys . . . . . . . . . . . . . . . . . 34
2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.1 Bayes . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.2 . . . . . . . . . . . . . . . . . 38
2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3 43
3.1 . . . . . . . . . . . . . . . . . . . . 43
3.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
iii
-
3.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 . . . . . . . . . . . . . 48
3.2.1 Bayes Bayes . . . . . . . . . . . . . . . . . . . 48
3.2.2 Bayes . . . . . . . . 49
3.3 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.1 . . . . . . . . . . . . . . . . . . . . . . . 52
3.3.2 . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.3 . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.1 . . . . . . . . . . . . . . . . . . . . 64
3.5.2 Bayes . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.7 . . . . . . . . . . . . . . . . 72
3.7.1 . . . . . . 72
3.7.2 . . . . . . . 74
3.8 . . . . . . . . . . . . . . . . . . . . . . . 78
3.8.1 . . . . . . . . . . . . . . . . . . . . 78
3.8.2 . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
89
.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 89
.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
95
96
97
-
f(x|)
( ). x, -
.
ET , E(T ) ( ) f(x
|). .. T = T (X
) X f(x
|)
ET E(T ) :=XT (x
)f(x
|)dx
.
L(d, ) . d .
R(, ) = (X),
R(, ) := EL(, ) =
XL((x
), )f(x
|)dx
.
() .
Epi{g()} (),
Epi{g()} :=
g()()d.
(|x) ,
(|x) :=
f(x|)()
f(x|)()d .
Epi{g()|x}
(|x),
Epi{g()|x
} :=
g()(|x
)d.
pi(d|x) d,
pi(d|x) := Epi{L(d, )|x
} =
L(d, )(|x
)d.
rpi() Bayes = (X),
rpi() :=
R(, )()d =
XL((x
), )f(x
|)()dx
d.
m(x) ,
m(x) :=
f(x
|)()d.
-
IA(x) A,
IA(x) :=
{1, x A,0, x / A.
exp(x), ex
ex =n=0
xn
n!, x R .
log(x), log x , exp(x).
(x)
(x) :=
0
tx1etdt, x > 0.
(1) = 1, (1/2) = (x + 1) = x(x). n N,
(n+ 1) = n!.
B(x, y)
B(x, y) :=
10tx1(1 t)y1dt, x, y > 0.
B(x, y) =(x)(y)
(x+ y), x, y > 0.
g(x) f(x) g f , c > 0 g(x) = cf(x), x.
-
1
1.1 Bayes
I Bayes
.
. . A, B
1 ,
P(A|B) = P(B|A)P(A)P(B)
. (1.1.1)
. ,
P(A|B) = P(A B)P(B)
P(B|A) = P(A B)P(A)
,
P(A B) = P(A|B)P(B) = P(B|A)P(A) P(A|B) (1.1.1). , , P(B) (1.1.1)
P(B) = P(B|A)P(A) + P(B|Ac)P(Ac)
( Ac A)
P(A|B) = P(B|A)P(A)P(B|A)P(A) + P(B|Ac)P(Ac) . (1.1.2)
1 (,A ,P),
A ,
P : A [0, 1] () Kolmogorov:
1. P(A) > 0, A A ,
2. P() = 1
3. A1, A2, . . . A (), P(Ai) =
P(Ai).
1
-
2 1.
, A1, A2, . . .
( ),
,
P(Ai|B) = P(B|A)P(Ai)P(B)
=P(B|Ai)P(Ai)j
P(B|Aj)P(Aj) . (1.1.3)
X, Y ( )
X , Y . (1.1.3) Ai = {Y = yi}, B = {X = x}, x X , yi Y,
P(Y = yi|X = x) = P(X = x|Y = yi)P(Y = yi)yY
P(X = x|Y = y)P(Y = y) .
fX|Y (x|y) = P(X = x|Y = y), fY |X(y|x) = P(Y = y|X = x), fX(x) =P(X = x) fY (y) = P(Y = y),
fY |X(yi|x) =fX|Y (x|yi)fY (yi)
fX(x)=
fX|Y (x|yi)fY (yi)yY
fX|Y (x|y)fY (y). (1.1.4)
(1.1.4) X, Y
:
fY |X(y|x) =fX|Y (x|y)fY (y)
fX(x)=
fX|Y (x|y)fY (y)yY fX|Y (x|y)fY (y)dy
. (1.1.5)
Bayes
.
Bayes (1.1.1) -
(1.1.2):
A P(A), - B P(A|B).
1.1.1. -
.
() .
0.999 0.005.
0.003.
;
.
A =
Ac =
B =
-
1.1. Bayes 3
, P(B|A) = 0.005, P(B|Ac) = 0.999, P(A) =0.997, P(Ac) = 0.003. Bayes,
P(A|B) = P(B|A)P(A)P(B|A)P(A) + P(B|Ac)P(Ac) =
0.005 0.9970.005 0.997 + 0.999 0.003 = 0.625.
.
0.997,
0.625.
Bayes (1.1.5).
.
1.1.2. [Bayes (1763) ]
1 .
y. n X
(
y). X, y;
.
U(0, 1),
y fY (y) = I(0,1)(y) ={
1, 0 < y < 1,0, .
, y, Ber-
noulli y.
U U(0, 1),
P(|y) = P( y)= P(U 6 y) =
y0du = y .
y, X n
Bernoulli y, X|y B(n, y) ()
fX|Y (x|y) =(n
x
)yx(1 y)nx , x = 0, 1, . . . , n.
X y (0, 1)
fX|Y (x|y)fY (y) 2
fX(x) =
10
(n
x
)yx(1 y)nxdy =
(n
x
)B(x+ 1, n x+ 1)
=
(n
x
)(x+ 1)(n x+ 1)
(n+ 2)=
n!
x!(n x)!x!(n x)!(n+ 1)!
=1
n+ 1, x = 0, 1, . . . , n,
2 vii.
-
4 1.
, ,
n . (1.1.5) -
y X = x
fY |X(y|x) =fX|Y (x|y)fY (y)
fX(x)
= (n+ 1)
(n
x
)yx(1 y)nx
=1
B(x+ 1, n x+ 1) yx(1 y)nx , 0 < y < 1,
, Beta(x+ 1, n x+ 1).
: 1.1, 1.2, 1.3, 1.8.
1.2
x= (x1, . . . , xn) -
X
= (X1, . . . ,Xn)
.
( ) ,
R, > 1. ( -, ) ( )
g() . -
x, -
X
.
, () X
-
.
f(x; ), .
( Bayes)
:
(a priori) (), , f(x
; ) X
.
()
( ).
():
() .
() ( ) x
f(x
|).
f(x; ),
. X,
-
1.2. 5
X
, f(x|)(). ,
,
m(x) :=
f(x
|)()d (1.2.6)
( m marginal = ),
X
= x ( Bayes)
(|x) :=
f(x|)()m(x
)
=f(x
|)()
f(x|)()d . (1.2.7)
(a posteriori) ,
.
,
(
) .
1.2.1. X
= (X1, . . . ,Xn) Poisson
P(), = (0,),
Xi f1(x|) = e x
x!IZ+(x) , i = 1, . . . , n.
f(x|) =
ni=1
f1(xi|) =ni=1
exi
xi!IZ+(xi) = e
n
xixi!
IZn+(x) .
G(, ) , > 0 ,
() =
()1eI(0,)() .
,
m(x) =
f(x
|)()d =
0
(en
xixi!
)(
()1e
)d
=
()xi!
0
+
xi1e(+n) d (1.2.8)
=
()xi!
(+
xi)
( + n)a+
xi, x Zn+ , (1.2.9)
( )
(|x) =
f(x|)()m(x
)
=
(
()xi!
+
xi1e(+n))/(
()xi!
(+
xi)
( + n)a+
xi
)
=( + n)a+
xi
(+
xi)+
xi1e(+n) , > 0,
, G( +xi, + n).
-
6 1.
1.2.1. () (1.2.8) (+
xi)/(+n)+
xi.
G(+xi, +n) 0
0
( + n)a+
xi
(+
xi)+
xi1e(+n)d = 1,
, .
() m(x) (1.2.9)
x1, . . . , xn. (),
() .
.
1.2.2. y
1.1.2, X| B(n, ), = (0, 1), , () U(0, 1). Beta(x+ 1, n x+ 1).
1.2.2. -
. -
. (
) . , =
{1, 2, . . .} m(x
) =
f(x|)() . (1.2.10)
-
m(x) =
f(x
|)(d) , (1.2.11)
[ ()d (d) ]
() (1.2.10) () . (
(1.2.11) Lebesgue.)
(Riemann)
,
.
: 1.4.
1.3
,
. , 50 60 !
,
.
-
1.3. 7
:
.
. () , -
.
.
, ()
.
:
;
-
. ,
. ,
( .. ) (
)
.
, -
,
. .
.
. 3
.
. , -
.
1.3.1.
I(X) = [T1(X
), T2(X
)] 100(1 )%
g()
P{g() I(X)} = P{T1(X
) 6 g() 6 T2(X
)} = 1 , .
100(1 )% - I(X
) = [T1(X
), T2(X
)] g(). , -
. x, T1(x
),
T2(x) 100(1 )% g()
I(x) = [T1(x
), T2(x
)], T1(x
) T2(x
).
I(x) g() 1; ! 1
I(X) g() ( ) I(x
)
g() .
3 http://www.mrc-bsu.cam.ac.uk/bugs BUGS (Bayesian inference Using Gibbs Sampling) WinBUGS Windows.
-
8 1.
1 I(x)
.
. x -
[L, U ]
Ppi{L 6 6 U |x
} =
UL
(|x)d = 1 ,
: [L, U ]
1 .
1.3.2. 4
p (pvalue).
p
( , -
) .
-
T (X) . T (x
),
p = PH0{T (X) > T (x
)},
, p
.5 -
p ( 0.05
).
:
-
; T (X) = T (x
)
T (x). ;
,
. -
,
:
4
.5 p .
: p !
-
1.3. 9
. x
L(|x), f(x
|)
x. , x
y
c (
)
L(|x) = cL(|y
) , , x
y
.
Bernoulli = (0, 1) H0 : = 1/2 H1 : > 1/2. 12
9 3 .
9 :
B(12, ) ( - 12 )
f1(x|) =(12
x
)x(1 )12xI{0,1,...,12}(x) ,
L1(|x = 9) =(12
9
)9(1 )3I(0,1)() = 2209(1 )3I(0,1)() ,
NB(3, 1 ) ( )
f2(x|) =(3 + x 1
x
)x(1 )3I{0,1,2,...}(x) ,
L2(|x = 9) =(11
9
)9(1 )3I(0,1)() = 559(1 )3I(0,1)() .
( ) ,
, -
,
.
x ( 1.5). X B(12, ) p
p1 = P=1/2(X > 9) =12x=9
f1(x|1/2) =12x=9
(12
x
)(1
2
)12= 0.0730
X NB(3, 1 ) p
p2 = P=1/2(X > 9) =
x=9
f2(x|1/2) =x=9
(2 + x
x
)(1
2
)3+x= 0.0327.
-
10 1.
, = 0.05
: -
( p2 < 0.05) ( p1 > 0.05).
.
,
10 12
10 . , -
.
1.4
, .
f(y) (-
) y. ,
Poisson P()
f(y) = ey
y!IZ+(y)
e y. -
P(). e
e ( ) .
, -
N (, 2),f(x) =
1
2
exp
{ 122
(x )2}IR(x) . (1.4.12)
f ( x) 1/(2). -
C 6= 1/(2)
g(x) = C exp
{ 122
(x )2}IR(x)
; !
R ,
1 =
xR
g(x)dx
=
xR
C2
2
exp
{ 122
(x )2}IR(x)dx
= C2
xR
f(x)dx
= C2,
-
1.4. 11
C = 1/(2).
( ) exp{ 122
(x )2}IR(x) (1.4.13)
N (, 2) 1/(2). N (, 2) (1.4.13) , .
f(x)
f(x) exp{ 122
(x )2}IR(x) .
( , .) . ,
exp
{ 122
(x )2}
= exp
{ x
2
22+x
2
}exp
{
2
22
}
exp{2/(22)} () x,
f(x) exp{ x
2
22+x
2
}IR(x) .
.
:
1.4.1. f : X R
> C1 :=
Xf(x)dx, X
xX
f(x) , X .
f(x) := Cf(x) X f(x), C f .
1.1 .
: 1.6.
1.2
(|x) =
f(x|)()m(x
)
.
-
12 1.
. . .
Poisson,P() f(x) = e xx! IZ+(x) x
x! IZ+(x)
,G(, ) f(x) = () xa1exI(0,)(x) xa1exI(0,)(x)
,Beta(, ) f(x) = (+)()() xa1(1 x)1I(0,1)(x) xa1(1 x)1I(0,1)(x)
1.1: ( x).
,
m(x) x
.
(|x) f(x
|)() ,
C = 1/m(x).
m(x) =
f(x
|)()d .
m(x), .
1.4.1.
: f .
f1(x) exp{ 122
(x )2}IR(x) ,
f2(x) exp{ 122
(x )2}I(0,)(x) .
f1 N (, 2) ( - C = 1/
2 ) f2 (0,).
f2 :
C1 =
x=0
exp
{ 122
(x )2}dx
=
y=/
exp{y2/2}dy [ y = (x )/]
= 2{1 (/)}
= 2(/) ,
f2(x) =1
(/)2
exp
{ 122
(x )2}I(0,)(x) .
: 1.7.
-
1.5. 13
1.5
I
.
.
1.5.1. [ ]
X f(x
|), . T = T (X
) ( )
X
T = t t
.
,
. -
, .
. -
.
, -
. ,
.
. ,
-
,
. T = T (X) ,
Neyman Fisher
f(x|) = q(T (x
), )h(x
) , x
, ,
(|x) =
f(x|)()
f(x|)()d =
q(T (x), )h(x
)()
q(T (x), )h(x
)()d
=q(T (x
), )()
q(T (x), )()d
.
-
( ) .
1.5.1. 1.2.1 -
Poisson P(), = (0,), G(, ), G(+xi, +n). - T = T (X
) =
Xi P(n) (
Poisson Poisson
). ,
x t,
(|t) f(t|)() {ent
}{1eI(0,)()
}= +t1e(+n)I(0,)() ,
-
14 1.
G( + t, + n) .
1.6
1.1. , ,
.
, , ,
1/2. ,
( ):
.
.
,
p. , .
, .
9%
.
() p = 0.9, ( )
;
() (0, 1)
p,
.
1.2. = {1, 2} (1) = (2) = 1/2 X| = 1 N (1, 2) X| = 2 N (2, 2) > 0 .() X; ( )
2.
() P( = 1|X = 1) 2. 1/2
2, X = 1
{ = 1}.
1.3. [ Monty Hall ] -
, .
(). ,
.
( ) . ;
, ;
1.4. () -
X f(x
|), .
-
1.6. 15
(|x), Y
g(y
|), , X.
, (|x, y
)
:
() X, Y
(X, Y)
()
() () : X
(|x), Y
.
1.5. () X B(12, ), = (0, 1). = 0.05 H0 : = 1/2
H1 : > 1/2
1(X) =
1 [. H0] , X > 9,
0.5718 [. H0 0.5718] , X = 9,
0 [. ( ) H0] , X < 9.
() X NB(3, 1 ), =(0, 1). = 0.05
H0 : = 1/2 H1 : > 1/2
2(X) =
1 [. H0] , X > 8,
0.7867 [. H0 0.7867] , X = 8,
0 [. ( ) H0] , X < 8.
() X = 9
.
() 1.3.2 p p1 = 0.0730
p2 = 0.0327 .
1.6. .
() X f(x) exp{ax2 + bx}IR(x) a > 0, X N (b/2a, 1/2a).() Y f(y) yaeby I(0,)(y) a > 1, b > 0 Y G(a+ 1, b).() X f(x) (1 x)bI(0,1)(x) b > 1, X Beta(1, b+ 1).() X f(x) (ax/x!)IZ+(x) a > 0, X P(a).() Y f(y) I(a,b)(y), Y U(a, b).() Y f(y) ayIZ+(y) 0 < a < 1, Y Ge(1 a).1.7. -
. ( .)
() f(y) (y/y!)I{1,2,...}(y), > 0.() f(x) axI{0,1,...,n}(x), a > 0.() g(x) exp
{ (x)222
}I(a,b)(x), 6 a < b 6.
() g(z) ezI(,)(z), > 0, > .
-
16 1.
() f(x) [x(1 x)]1/2I(0,1)(x).() f(x) 1
1+(x)2I(a,b)(x), R, 6 a < b 6.
.
1.8.* [Lindley (1965) ]
.
x X. xx
, xX Xx, XX .
p2 2p(1p), 0 < p < 1 .
x X 1/2 .
() ,
2p/(1 + 2p)
.
()
n , .
;
-
2
. ()
. ( ) !
.
.
, (
,
).
2.1
. -
, ( )
1.
,
. ,
, ,
.
,
.
.
,
.
1 .
17
-
18 2.
(
), ,
. (
.) ,
,
, ,
, .
.
( )
, .
2.2
2.2.1
.
2.2.1. [ ]
A, B P(B)/P(A) B
A.
B A
B , A.
,
= {1, 2}. . :
; .. 1 2 ,
,
(1)
(2)= 2 2(2) = (1) = 1 (2) 3(2) = 1,
(2) = 1/3 (1) = 2/3.
,
(1) = (2) = 1/2 ().
-
. = {1, 2, . . . , } ( ) -
.
: .
2.2.1. = {1, 2, 3}. 1, 2 3.
-
2.2. 19
1
2 2 3. , 1
3.
. (1)/(2)
= 2 (2)/(3) = 3,
(1)
(3)=
(1)
(2) (2)(3)
= 2 3 = 6 6= 5
.
.
() -
. (
.) ,
1, 2 (2)/(1)
.
: 2.1.
2.2.2
() , =
[a, b] (a, b) [a, b) (a, b], < a < b < . , ()
. k > 1
1, . . . , k a < 1 < . . . < k < b, k + 1
[a, 1], . . ., (k, b].
2.2.2. -
I: , = (0, 1).
= (0, 0.25) [0.25, 0.50) [0.50, 0.75) [0.75, 1) =: 1 2 3 4 . ()
3 4,
2 1. , 3
4, 2 1. (i) := P( i),
(3) = 2(4) = 3(2) = 6(1) ,
, (1),
1 =
4i=1
(i) = (1 + 2 + 6 + 3)(1) (1) = 1/12
-
20 2.
0 0.25 0.50 0.75 1
1/12
2/12
3/12
6/12
2.1: 2.2.2 .
(2) = 2/12, (3) = 6/12, (4) = 3/12. ( )
2.1.
, -
. -
( ) .
,
( ) . (
.)
-
() , , a =
0 < 1 < . . . < k < k+1 = b ,
( ). ,
() = 1 ( ),
(i, (i))
( ) .
.
2.2.3. 2.2.2 -
0, 0.25, 0.5, 0.75, 1. -
0 0.75
. (0.75)/(0) = 10.
0.5 1
0.75, (0.5) = (1) = (0.75)/2.
0.25 0, (0.25) = 2(0). (0) = 1,
(0.25) = 2, (0.5) = (1) = 5 (0.75) = 10.
2.2()
(0, 1), (0.25, 2), (0.5, 5), (0.75, 10), (1, 5) 5,
-
2.2. 21
0 0.25 0.50 0.75 1
12
5
10
0 0.25 0.50 0.75 1
0.20.4
1.0
2.0
() ()
2.2: () () ( - ) - 2.2.3.
2.2().2
,
( ) , .. 0 0.5
.
.
. (
) , 0
. , e 1/2; ( .)
. (
.)
: 2.2.
2.2.3
:
-
(.. , , )
2 .
() ( ).
-
22 2.
.
, , .
2.2.4. 100
20. = R,
. ,
: N (100, 202).
2.2.5. = (0, 1) - .
Beta(, ), , > 0. , ( )
0.7 0.05.
0.7 =
+
0.05 =
(+ )2(+ + 1)
, ,
Beta(, ) ( ). = 2.24, = 0.96, Beta(2.24, 0.96). ( 2.3.)
,
. , 95%
0.4, (0.4, 1).
,
0.7 =
+
0.95 =
10.4
(+ )
()()1(1 )1d = Ppi( > 0.4)
, . 3
( ) = 4.83,
= 2.07, Beta(4.83, 2.07).
: 2.4.
. -
.
3 , ,
. , ,
, .
-
2.3. 23
2.2.6. [Berger (1985) ]
0 1 () 1 (). N (, 2), . = 0,
. , z > 0
Ppi( 6 z) = Ppi( > z). z = 1
Ppi( 6 1) = Ppi( > 1) = 0.25 Ppi
( 0
>1 0
)= 0.25 P(Z > 1/) = 0.25
Z N (0, 1). 1/ = z0.25 0.675 (z ), 2 2.19. , N (0, 2.19).
Cauchy C(, ) -
() =1
1
1 + [( )/]2 IR() ,
= 0 = 1.
C(, ) ( )
() =
1
1
1 + (y/)2dy =
1
2+
1
arctan
(
), R.
, () (1) = 1/4, (1) = 3/4 arctan(1/) = /4, = 1 tan(/4) = 1.
2.3
.
2.3.1. [ ]
F f(x
|) F ,
F . f(x
|) f(x
|).
f(x|) .
.
.. ,
.
-
24
2.
f(x|) pi() pi(|x)
B(n, ) Beta(, ) Beta(+ x, + n x)
x(1 )nx 1(1 )1 +x1(1 )+nx1
Poisson P(n) G(, ) G(+ x, + n)
enx 1e +x1e(n+)
N (, 2/n) ( ) N (, 2) N(n2x+2n2+2
, 22
n2+2
) exp{ n
22(x )2} exp{ 1
22( )2} exp{n2+2
222( n2x+2
n2+2)2}
N (, ) ( ) IG(, ) IG(+ n/2, +(xi )2/2) n/2 exp{ 12(xi )2} (+1)e/ (+n/2+1) exp{1 [ + 12(xi )2]} G(n, ) G(, ) G(+ n, + x)
nex 1e +n1e(+x)
2.1: .
-
2.3. 25
,
.
2.3.1. [Poisson ] 1.2.1
X
= (X1, . . . ,Xn) Poisson P(), = (0,),
f(x|) = en
nxxi!
IZn+(x)
() G(, ), (|x) G( + nx, + n).
.
Poisson.
, +nx, +n
.
: 2.5.
2.3.2. [ ] X B(n, ), = (0, 1),
f(x|) =(n
x
)x(1 )nxI{0,1,...,n}(x) .
f(x|) , ( ) (1 ) . . , () Beta(, ) , > 0,
() 1(1 )1I(0,1)() ,
(|x) f(x|)() +x1(1 )+nx1I(0,1)() Beta(+ x, + n x) ., .
2.3.1. () X B(n, ) X1, . . . ,Xn Ber-
noulli B(1, ). ni=1Xi B(n, ) , X := ni=1Xi - .
() U(0, 1) : Beta(1, 1) , 1.1.2 Bayes .
2.3.3. [ ( ) ] X
= (X1, . . . ,Xn)
N (, 2), = R, > 0 .
Xi f1(x|) = 12
exp
{ 122
(x )2}IR(x) , i = 1, . . . , n,
-
26 2.
f(x|) =
ni=1
f1(xi|) =ni=1
1
2
exp
{ 122
(xi )2}
=1
n(2)n/2exp
{ 122
(xi )2
}
=1
n(2)n/2exp
{ 122
[(xi x)2 + n(x )2
]}
exp{ n22
(x )2}, R .
ni=1
(xi a)2 =ni=1
(xi x)2 + n(x a)2 .
N (, 2),
() =1
2
exp
{ 122
( )2}, R .
,
(|x) f(x
|)()
exp{ n22
(x )2}exp
{ 122
( )2}
= exp
{nx
2
22 n
2
22+nx
2
2
22
2
22+
2
}
exp{12
(n
2+
1
2
)2 +
(nx2
+
2
)
}
= exp
{12
(n2 + 2
22
)[2 2 n
2x+ 2
n2 + 2
]}
exp{12
(n2 + 2
22
)[2 2 n
2x+ 2
n2 + 2+
(n2x+ 2
n2 + 2
)2]}
= exp
{12
(n2 + 2
22
)( n
2x+ 2
n2 + 2
)2}, R .
(|x)
(n2x+ 2)/(n2 + 2) 22/(n2 + 2),
|x N
(n2x+ 2
n2 + 2,
22
n2 + 2
).
,
.
-
2.3. 27
40 160100 110.39
()
(|x = 115)
2.3: N (100, 225) N (110.39, 69.23) 2.3.4.
2.3.2. X N (, 2) ( )
2/n. 2.3.3
X N (, 2) ( ) 2/n 2.
2.3.4. [Berger (1985) ] . -
X
N (, 100), IQ . (, IQ, .) -
,
N (100, 225). x. 2.3.3 x
Epi{|x} = 100 100 + 225x
100 + 225=
400 + 9x
13
Varpi{|x} = 100 225
100 + 225= 69.23.
x = 115 , IQ , ,
N (110.39, 69.23). ( (400 + 9 115)/13 = 110.39.) 2.3.5. [ ] X G(, ), = (0,) > 0,
f(x|) =
()x1exI(0,)(x) .
f(x|) , ( ) e .
. , () G(, ) , > 0,
() 1eI(0,)() ,
-
28 2.
(|x) f(x|)() +1e(+x)I(0,)() G(+ , + x) .
, .
2.3.3. X G(, ) > 0 ,
X
= (X1, . . . ,Xn) -
E() ( EXi = 1/) X :=
ni=1Xi G(, ) = n
X1, . . . ,Xn Xi G(i, ) 1, . . . , n , - X :=
ni=1Xi G(, ) =
ni=1 i.
2.3.6. [ ( ) ]
X
= (X1, . . . ,Xn) N (, ), = (0,), R .
f(x|) = 1
(2)n/2exp
{ 12
(xi )2
}IRn(x
) .
IG(, ), , > 0,
() =
()
1
+1e/I(0,)()
( ).
,
1/ G(, ). ( !)
(|x) f(x
|)() 1
+n/2+1exp
{1
( +
(xi )2
2
)}I(0,)() ,
IG( + n/2, +(xi )2/2). .
: 2.6, 2.7, 2.8.
2.4
-
.
.
.
-
2.4. 29
2.4.1. = (0, 1) - Bernoulli.
U(0, 1) () = 1, (0, 1), .
2.4.2. -
, = R. () N (0, 1010) ,
0 1010
. ( .)
.. = (0,) () G(0.001, 0.001). 0.001/0.001 = 1 0.001/0.0012 = 1000.
.
2.4.1 ( )
,
()
()d = + .
()
. , () A
(A) :=
A()d
-
30 2.
2.4.1. [ ( ) ]
()d = + ,
, -
. x
m(x),
m(x) =
f(x
|)()d
-
2.4. 31
=
(n
x
) 10x1(1 )nx1d.
B(x, nx) ( vii) x 6= 0, n, . , x 6= 0, n,
m(x) =
(n
x
)B(x, n x) = n!(x)(n x)
x!(n x)!(n) =n!(x 1)!(n x 1)!x!(n x)!(n 1)! =
n
x(n x) 0 ).
= R. -
f(x ) X Y = X + c, c R . Y f(y c), , = + c, f(y), - Y . X| f(x ) Y | f(y ), X Y ,
-
32 2.
.
() ()
, A R,
Ppi( A) = Ppi( A) . (2.4.2)
= + c,
Ppi( A) = Ppi( + c A) = Ppi( A c) , (2.4.3)
A c := {a c : a A}. (2.4.2) (2.4.3)
Ppi( A) = Ppi( A c) , c R , A R . (2.4.4)
2.4.3. (2.4.4) -
.
(2.4.4) A()d =
Ac
()d =
A( c)d, c R , A R ,
() = ( c) , c, R .
= c,
(c) = (0) , c R , .
. :
2.4.1. = R,
() = 1.
() = 1 ()d =
d = .
(,).
-
2.4. 33
2.4.4. [ ]
X
f(x|) = 1f(x
),
( x/ )
.
2.4.7.
, N (0, 2), t, Tm(0, 2), G(, ) ( ), U(0, ).
= (0,). 1f(x/) -
X Y = cX c > 0 . Y
(c)1f(y/(c)), , = c, 1f(y/),
Y . X| 1f(x/) Y | 1f(y/), X Y
,
.
() ()
, A (0,),
Ppi( A) = Ppi( A) . (2.4.5)
= c,
Ppi( A) = Ppi(c A) = Ppi( c1A) , (2.4.6)
c1A := {a/c : a A}. (2.4.5) (2.4.6)
Ppi( A) = Ppi( c1A) , c > 0, A (0,) . (2.4.7)
2.4.5. (2.4.7) -
.
(2.4.7) A()d =
c1A
()d = c1A(/c)d, c > 0, A (0,) ,
-
34 2.
() = c1(/c) , c, (0,) .
= c,
(c) = c1(1) 1/c, c > 0.
(1) , :
2.4.2. = (0,), -
() =1
.
()d =
0
1
d = .
, -
.
.
.
2.4.8. 2.4.1 -
( ) = (0, 1) U(0, 1) . - logit,
,
logit() := log
(
1 ).
() = I(0,1)() = logit() R, = e/(1 + e) d/d =e/(1 + e)2
logit() = () = e
(1 + e)2IR() .
( .) () .
2.4.3 Jeffreys
1946 Jeffreys
. ,
-
2.4. 35
Fisher4,
I() = E
{[
log f(X
|)]2}
, (2.4.8)
I() = E{2
2log f(X
|)}. (2.4.9)
Jeffreys
J() I() . (2.4.10)
J() -
. , = g() ,
I() = E
{[
log f(X
|)]2}
= Eg()
{[
log f(X
|g())
]2}( g )
= Eg()
{[
g()log f(X
|g())
]2}[g()]2 ( )
= E
{[
log f(X
|)]2}(d
d
)2( = g())
= I()
(d
d
)2,
I() = I()
(d
d
)2= I(g1())
(d
d
)2. (2.4.11)
, = g()
() = J(g1())
dd I()
dd =I() ,
(2.4.11). J()
Fisher .
J()
. .
4 Fisher I: .
CramerRao .
-
36 2.
2.4.9. X
= (X1, . . . ,Xn) Bernoulli
B(1, ), = (0, 1). f(x
|) =
xi(1 )n
xiI{0,1}n(x
) ,
log f(x|) = xi log + (nxi) log(1 ) , x
{0, 1}n ,
log f(x
|) =
xi
n
xi1 , x {0, 1}
n ,
2
2 log f(x|) =
xi
2 n
xi
(1 )2 , x {0, 1}n ,
Fisher
I() = E{2
2log f(X
|)}
= E
{Xi2
+nXi(1 )2
}=
n
2+
n n(1 )2 =
n
(1 ) .
, , Jeffreys
J() I() =
n
(1 ) 1/2(1 )1/2 , 0 < < 1,
Beta(1/2, 1/2). 2.4.10. X
= (X1, . . . ,Xn)
N (, 2), = R, > 0 .
f(x|) = 1
n(2)n/2e
1
22
(xi)
2
IRn(x) ,
log f(x|) = n log n
2log 2 1
22
(xi )2 , x Rn ,
log f(x
|) = (xi )/2 , x
Rn ,
2
2 log f(x|) = n/2 , x
Rn ,
Fisher
I() = E{2
2log f(X
|)}
= n/2 ,
. , ,
Jeffreys
J() I() =
n/ 1, R .
Jeffreys -
.
: 2.10, 2.11, 2.12.
= (1, . . . , ) R( > 1), Jeffreys
Fisher,
I() =
(E
{2
ijlog f(X
|)})
.
-
2.5. 37
2.4.11. X
= (X1, . . . ,Xn)
N (, 2), = (1, 2) := (, ) = R (0,) ( ).
f(x|) = 1
n(2)n/2exp
{ 122
(xi )2
}IRn(x
) ,
log f(x|) = n log n
2log 2 1
22
(xi )2 , x Rn ,
log f(x
|) = (xi )/2 ,
2
2log f(x
|) = n/2 ,
log f(x
|) = n
+
1
3
(xi )2 ,2
2 log f(x|) = n
2 34
(xi )2 ,2
log f(x|) = 2(xi )/3 .
Fisher
I() = I(, ) =
E
{2
2log f(X
|)}
E{
2
log f(X|)}
E{
2
log f(X|)}
E{
2
2 log f(X|)}
=
n
20
0 n2
+3n
2
=
n
20
02n
2
( E
(xi ) = 0 E
(xi )2 = n2 ), Jeffreys = (, )
J() = J(, ) = |I()|1/2 =
2n2
4=
n2
2 1
2.
J(, ) = 1/2.
.
Jeffreys
. ,
1, . . . , .
2.4.12. 2.4.11
Jeffreys = (, ) J(, ) = 1/2.
(, ) = 1/, -
() = 1 () = 1/ ( , ,
). 3.7 .
-
38 2.
2.5
2.5.1 Bayes
Bayes
( ).
= (1, . . . , ) H R, > 1 (|). ( ) . , G(, ), =(, ) H = (0,)2 = H =(0,) .
x= (x1, . . . , xn) .
m(x|) =
f(x
|)(|)d.
Bayes :
.
, .
2.5.1. X
= (X1, . . . ,Xn) Poisson P(), = (0,) G(, ) > 0 = H = (0,) . 1.2.1 (1.2.9)
m(x|) =
()xi!
(+
xi)
( + n)a+
xiIZn
+(x) .
( !) x6= (0, . . . , 0), m(x
|)
= n/
xi = /x.
Bayes,
G(,/x).
: 2.13.
2.5.2
.
.
Bayes
-
2.6. 39
().
, , (|)(). ,
,
() =
H(|)()d ,
(
) . ,
.
2.5.2. 2.5.1
() G(, ), , > 0 . ,
(, ) = (|)() ={1e
()I(0,)()
}{1e
()I(0,)()
},
:
() =
(, )d 1I(0,)()
0
+1e(+) d =(+ )
( + )+1I(0,) .
()/() F2,2 ( ).
(|x)
{en
xi}{
1
( + )+
}I(0,)()
=+
xi1en
( + )+I(0,)() ,
.
2.5.2 -
.
.
.
.
: 2.14.
2.6
2.1. = {1, 2, . . . , 6}. 6 , 2 5
.
-
40 2.
() 2 1
;
() (); ( .)
2.2.
.
.
. = (0,), - max
= (0, max).
.
2.3. Beta(, ), , > 0.() Epi() = , Varpi() = 2,
=2(1 )
2 = (1 )
2
2 (1 ) .
2
;
() 2.2.5,
= 2.24, = 0.96.
2.4. G(, ), , > 0. () Epi() = , Varpi() = 2, = 2/2 = /2 .
()
5 2.
2.5. X
= (X1, . . . ,Xn)
Xi P(ti), i = 1, . . . , n, = (0,) t1, . . . , tn . G(, ), , > 0, .
2.6. X
= (X1, . . . ,Xn) U(0, ), = (0,) Pareto Par(I)(, ),, > 0,
() =
+1I[,)() .
Par(I)(+ n,max{, x(n)}), x(n) :=max{x1, . . . , xn}, Pareto U(0, ).2.7. X
(
), A(), B(x), C() T (x
)
f(x|) = exp{A() +B(x
) + C()T (x
)}, x
, ,
-
2.6. 41
X := {x: f(x
|) > 0} .
(|, ) exp {A() + C()} I() , , . f(x
|)
.
2.8. F f(x|).
n = 2, 3, . . .
Fn :={
ni=1
pii
1, . . . , n F 0 < p1, . . . , pn < 1,ni=1
pi = 1
}.
n Fn f(x|).
2.9. () (
) m(x) < . ( A X
A f(x|)dx
> 0 m(x
) =.)
2.10. X
= (X1, . . . ,Xn) E(), = (0,).() ;
() ()
.
() Jeffreys
. ;
2.11. X
= (X1, . . . ,Xn) Poisson P(), = (0,). Jeffreys
. ;
2.12. X
= (X1, . . . ,Xn) NB(, ), = (0, 1), > 0 .() -
. () Beta(, ) ;() Jeffreys
. ;
2.13. X B(n, ), = (0, 1), () Beta(, 1), > 0, . X, m(x|), .
x;
2.14. X N (, 2), = R, R 2 > 0.() .
;
() N (0, 1). ;
-
3
3.1
3.1.1
() , x
X f(x
|),
R, > 1. d,
D. D ,
.
g(), g ( ) ,
a g() =: D g. H0
H1 d H0,
D = {0, 1} 1 0 .
3.1.2
, (
D) : ,d, ( )
.
L : D R L(d, ) d
.1 d
, .
1
.
43
-
44 3.
3.1.1. [ ]
L : D R 1. L(d, ) > 0, d, ,2. L(d, ) = 0, d , .
3.1.1. () g(), g
.
L(d, ) = {d g()}2 , (3.1.1)
L(d, ) = w(){d g()}2 , (3.1.2)
w() > 0,
L(d, ) = |d g()| . (3.1.3)
d = g(),
1 2 3.1.1.
: 3.1.
() H0 : 0 H1 : 1 D = {0, 1} ( 0 1 H0).
01
L(d, ) =
{d, 0 ,1 d, / 0 , (3.1.4)
0 1 .
01 I II2 (
1).
01,
L(d, ) =
{0d, 0 ,1(1 d) , / 0 , (3.1.5)
0 1 . 0
, 0 I 1 II.
2 II: .
I H0 II H0 .
-
3.1. 45
3.1.3
.
3.1.2. [ ]
3
D.
g()
. ()
.
= (X)
d = (x), x
.
3.1.2. g()
() = (X) = X x
= (x1, x2) = (2.5, 3),
( g()) d = (x) = x = (x1 + x2)/2 =
(2.5 + 3)/2 = 0.25.
3.1.4
-
.
3.1.3. [ ]
L(d, ) = (X) ,
R(, ) := EL((X), ) =
XL((x
), )f(x
|)dx
(3.1.6)
.
( .)
3.1.3. () g().
(3.1.1)
I
R(, ) (, ) = E{ g()}2 . (3.1.7)
(3.1.2) -
,
R(, ) = w()(d, ) = w()E{ g()}2 , (3.1.8)3 .
-
46 3.
(3.1.3)
R(, ) = E| g()| . (3.1.9)
() H0 : 0 H1 : 1 01 (3.1.4).
H0 X C X ,
C = C(X) =
{1, X
C
0, X
/ C
( C ). 0
R(C , ) = 1 P(X C) + 0 P(X
/ C) = P(X
C) = P( I) ,
1
R(C , ) = 0 P(X C) + 1 P(X
/ C) = P(X
/ C) = P( II) .
3.1.4. R(, ) 1, 2 ,
1 2
R(1, ) 6 R(2, ) , ,R(1, 0) < R(2, 0) , 0 .
3.1.1. 3.1.4
1 2
( 0)
. 6 =. ,
g() X1,X2
f1(x|), 1 = 1(X1,X2) = X1 2 = 2(X1,X2) = X2 g()
R(1, ) =
XL(x, )f1(x|)dx = R(2, ) , .
R(1, ) 6 R(2, ), , 1 2. ( .)
-
3.1. 47
3.1.5. [ ]
:
() = (X)
.
() = (X)
.
3.1.2. -
. ,
,
-
. ( I
g() .)
: 3.2, 3.3.
3.1.5
. -
.
.
, ,
x , (x
), ,
.
-
, :
;
. ,
D, D.
.
.
; (
.)
( .. 4)
4 = (X) g() E = g(), .
-
48 3.
. ,
,
.
3.2
, -
f(x|) ()
x
(|x) =
f(x|)()m(x
)
=f(x
|)()
f(x|)()d .
1 2
L(d, ).
, 1 2 = c1,
R(1, ) < R(2, ) , 1 ,R(1, ) > R(2, ) , 2 .
( ,
.) 2 ( 1). ,
1 -
( ) . ,
2,
1.
3.2.1 Bayes Bayes
()
. -
( )
, (
) ,
-
. , ,
Bayes
( ) .
-
3.2. 49
3.2.1. [ Bayes ]
rpi() :=
R(, )()d =
XL((x
), )f(x
|)()dx
d (3.2.10)
Bayes .
rpi() (
x). , , , ,
Bayes , rpi(). , 1, 2
rpi(1) < rpi(2)
1 . , -
. (
L(d, ) ())
Bayes.
3.2.2. [ Bayes ]
Bayes (3.2.10)
Bayes ( ).
3.2.2 Bayes
,
. ,
x ,
x X . (
) .
3.2.3. [ ]
pi(d|x) := Epi{L(d, )|x
} =
L(d, )(|x
)d (3.2.11)
d.
pi(d|x) d, x
: -
.
pi(d|x) d D.
-
50 3.
3.2.4. [ Bayes ]
pi(x) D pi(d|x
)
Bayes (
).
5
( ) Bayes.
, Bayes
Bayes.
3.2.1. Bayes ,
Bayes ( )
pi = pi(X), x
pi(x
)
Bayes.
.
(|x)m(x
) = f(x
|)() , x
, .
Bayes = (X)
rpi() =
XL((x
), )f(x
|)()dx
d
=
XL((x
), )(|x
)m(x
)dx
d
=
X
(L((x
), )(|x
)d
)m(x
)dx
=
Xpi((x
)|x)m(x
)dx
>
Xpi(pi(x
)|x)m(x
)dx
(3.2.12)
= rpi(pi) .
( Fubini,
, L(d, ) > 0, d, .)
3.2.1. (3.2.12) pi(x)
Bayes, = (X)
pi(pi(x)|x) 6 pi((x
)|x) , x
.
5 ( ) .
-
3.3. Bayes 51
(3.2.12) .
Bayes ( ;), Bayes
. ()
.
3.2.5. [ Bayes ]
() ,
pi = pi(X), x
pi(x
) Bayes, -
Bayes.
Bayes . ,
-
Bayes
Bayes.
.
3.2.2. Bayes
( ).
. pi Bayes. , , -
Bayes , ,
rpi(pi) < rpi() . (3.2.13)
. pi
.
pi,
R(, ) 6 R(pi, ) , ,
()
rpi() =
R(, )()d 6
R(pi, )()d = rpi(
pi)
(3.2.13).
pi, .
3.2.2 Bayes
, .
-
52 3.
3.3 Bayes
g() g : g() =: D -, . ,
g(),
g().
3.3.1. [ Bayes ]
Bayes g() Bayes
g() ( ).
3.3.1
L(d, ) = {d g()}2.
3.3.1. L(d, ) = {d g()}2, Bayes g() g() ,
pi(x) = Epi{g()|x
}.
, Bayes g()
pi = pi(X) = Epi{g()|X
}.
. () d
pi(d|x) = Epi{d g()|x
}2 = d2 2dEpi{g()|x
}+ Epi{g()2|x
}.
d 6 d = Epi{g()|x},
Bayes pi(x).
: 3.4.
3.3.1. X
= (X1, . . . ,Xn)
N (, 2), = R, > 0 , N (, 2). 2.3.3
(|x) N
(n2x+ 2
n2 + 2,
22
n2 + 2
).
6 x2+x+, 6= 0, /(2). > 0 < 0.
-
3.3. Bayes 53
, 3.3.1, Bayes
pi = pi(X) =
n2X + 2
n2 + 2
Bayes (n2x +
2)/(n2 + 2).
3.3.2. 2.3.4
x = 115 IQ, IQ
N (110.39, 69.23). , IQ 110.39
.
3.3.1. Bayes
pi = pi(X) =
n2
n2 + 2X +
2
n2 + 2
7 X, (
I X
, .)
, ,
. , Bayes, pi(x), , x,
, . : n,
pi(x) x, ,
.
3.3.3. n -
N (, 2), = R, > 0 , () = 1. ,
(|x) f(x
|)() exp
{ n22
( x)2}IR() ,
N (x, 2/n). x, ,
, X ( ) -
Bayes . (,
.) ( )
X .
: 3.5.
7 a, b R, > 1, c [0, 1] ca + (1 c)b a, b. a, b .
-
54 3.
3.3.4. X B(n, ), = (0, 1), () Beta(, ). 2.3.2
(|x) Beta(+ x, + n x) .
Epi{|x} = (+ x)/(+ + n), , , Bayes .
Bayes
pi = pi(X) =+X
+ + n=
n
+ + n
X
n+
+
+ + n
+ .
Bayes -
, X/n ( X n
Bernoulli, ),
/( + ) ( ).
(1 ) ( L(d, ) ={d(1)}2), Bayes ,
Epi{(1 )|x} =
(1 )(|x)d
=
10(1 ) (+ + n)
(+ x)( + n x) +x1(1 )+nx1d
=(+ + n)
(+ x)( + n x) 10(+x+1)1(1 )(+nx+1)1d
=(+ + n)
(+ x)( + n x)(+ + n+ 2)
(+ x+ 1)( + n x+ 1)=
(+ x)( + n x)(+ + n)(+ + n+ 1)
.
: 3.6.
3.3.5.
Epi() = 0.7
Varpi() = 0.1.
5/8, 8/15, 5/10, 6/8 10/12 .
.
. (
.) 2.3, -
Beta(14, 6). X1, . . . ,X5 , Xi B(ni, ), i = 1, . . . , 5. n1, . . . , n5
-
3.3. Bayes 55
10
()(|x)
3.1: Beta(14, 6) Beta(48, 25) 3.3.5.
: n1 = 8, n2 = 15, .
f(x|) =
5i=1
fi(xi|) =5
i=1
(nixi
)xi(1 )nixi I{0,1,...,ni}(xi)
=
{5
i=1
(nixi
)I{0,1,...,ni}(xi)
}
xi(1 )
ni
xi , (0, 1) .
ni = 8 + 15 + 10 + 8 + 12 = 53
xi = 5 + 8 + 5 + 6 + 10 = 34.
(|x) f(x
|)()
{34(1 )5334} {141(1 )61} I(0,1)()= 481(1 )251I(0,1)() ,
Beta(48, 25). , Bayes Epi{|x
} =
48/(48 + 25) = 0.6575.
: 3.7.
3.3.6. X
= (X1, . . . ,Xn) Poisson P(), = (0,), () G(, ). 2.3.1
(|x) G( +xi, + n) = G(+ nx, + n) .
Epi{|x} = (+ nx)/( + n), ,
, Bayes .
Bayes
pi = pi(X) =
+ nX
+ n=
n
+ nX +
+ n
.
-
56 3.
1 2 3 4 5 6 7 8 9 10
5 1 5 14 3 19 1 1 4 22
94.32 15.72 62.88 125.76 5.24 31.44 1.05 1.05 2.10 10.48
3.1:
. (: Gaver and O Muircheartaigh, 1987.)
, Bayes
, X,
/ ( ).
2 ( L(d, ) =
{d 2}2), Bayes ,
Epi{2|x
} = (+ nx)(+ nx+ 1)
( + n)2.
( Epi{2|x}
2(|x
)d, EY 2 = VarY + (EY )2.)
: 3.8, 3.9.
3.3.7.
.
i ti, -
Xi P(ti), = (0,), X1, . . . ,X10 . 3.1.
() G(1.8, 0.9)..
f(x|) =
10i=1
fi(xi|) =10i=1
eti(ti)
xi
xi!IZ+(xi)
=
{10i=1
txii IZ+(xi)
}
xie
ti , (0,) .
xi = 5 + 1 + . . . + 22 = 75
ti = 94.32 + 15.72 + . . . + 10.48 = 350.04.
(|x) f(x
|)()
{75e350.04
}{1.81e0.9
}I(0,)()
= 76.81e350.94 I(0,)() ,
G(76.8, 350.94). - , Bayes ,
Epi{|x} = 76.8/350.94 = 0.2188.
: 3.10.
-
3.3. Bayes 57
3.3.2
3.3.2. L(d, ) = |d g()|, Bayes g() g()
.
. , .
M
P(Y 6M) > 1/2 P(Y >M) > 1/2
, ( ),
P(Y < M) 6 1/2 P(Y > M) 6 1/2
Y . ,
.
M g() d < M
(). ,
L(d, ) L(M,) = |d g()| |M g()| =
dM , g() 6 d2g() dM , d < g() < MM d, g() >M .
L(d, ) L(M,) > (dM)I(,M)(g()) + (M d)I[M,)(g()) .
,
pi(d|x) pi(M |x
) > (dM)Epi{I(,M)(g())|x
}+ (M d)Epi{I[M,)(g())|x
}
= (dM)Ppi{g() < M |x}+ (M d)Ppi{g() >M |x
}
= (M d)(Ppi{g() >M |x} Ppi{g() < M |x
})
> 0,
,
Ppi{g() >M |x
} > 1/2 > Ppi{g() < M |x
}.
d > M .
(): 3.11.
,
. Bayes
.
-
58 3.
3.3.8. X N (, 2/n), = R, > 0 () N (, 2), (|x) Epi{|x
} = (n2x +
2)/(n2+2).
. Bayes
Epi{|x} .
3.3.3
3.3.2. [ ]
(x) -
(|x). = (X
)
.
.
.
(|x)
, .
Bayes.
,
.
(|x) .
3.3.9. X B(n, ), =[0, 1], () Beta(1/2, 1/2). (|x
) Beta(x+ 1/2, n x+ 1/2).
x = 0 (|x = 0) 0.
x = n -
1.
, n > 2, 0 < x < n
(x 1/2)/(n 1).
= (X) =
0, X = 0,
X 1/2n 1 , 0 < X < n,
1, X = n.
: 3.14, 3.15.
-
3.4. 59
3.4
,
(credible set).
3.4.1. [ ]
C 100(1 )%
Ppi{ C|x
} =
C
(|x)d > 1 .
1
C Ppi{ C|x} = 1 .
( )
.
3.4.1. X
= (X1, . . . ,Xn)
N (, 2), = R, > 0 , () N (, 2).
(|x) N
(n2x+ 2
n2 + 2,
22
n2 + 2
),
Ppi
{n2x+ 2
n2 + 2 z/2
22
n2 + 26 6
n2x+ 2
n2 + 2+ z/2
22
n2 + 2
x}
= 1 ,
C :=
[n2x+ 2
n2 + 2 z/2
22
n2 + 2,n2x+ 2
n2 + 2+ z/2
22
n2 + 2
]
100(1 )% () . z ()
P(Z > z) = , Z N (0, 1). () =
1, (|x) N (x, 2/n),
Ppi
{x z/2
n6 6 x+ z/2
n
x
}= 1 ,
[x z/2
n, x+ z/2
n
]
100(1 )% .
-
60 3.
3.4.2. 2.3.4 x = 115
IQ, IQ N (110.39, 69.23)., 95% IQ
[110.39 1.9669.23 , 110.39 + 1.96
69.23] = [94.08, 126.70] .
( z/2 = z0.025 = 1.96.) 95%
IQ ( ).
(.. -
) .
:
3.4.2. [ ]
C 100(1)% 100(1 )%
C = { : (|x) > k},
k .
, .
( )
100(1)% [L , U ] . :
1. L = inf (
) U ,
Ppi{ > U |x
} = .
2. U = sup (
) L (1 ) ,
Ppi{ > L|x
} = 1 .
3. ( -
) L, U
UL
(|x)d = 1 (3.4.14)
-
3.4. 61
10.17080
2.5
5
7.5
10
Ppi( 6 0.1708|x = 0) = 0.95
10.0745 0.5795
2
4
Ppi(0.0745 6 6 0.5795|x = 3)
0.95
10.82920
2.5
5
7.5
10
Ppi( > 0.8292|x = 10) = 0.95
() () ()
3.2: 95% - 3.4.3.
(L|x) = (U |x
) . (3.4.15)
(3.4.14) Ppi{L 6 6 U |x} = 1 (3.4.15)
(|x) > k := (L|x
).
3.4.3. X B(10, ), =(0, 1), () Beta(1/2, 1/2) ( Jeffreys). x , Beta(0.5+x, 10.5x).
x = 0. ,
Beta(0.5, 10.5)
(|x = 0) = (11)(0.5)(10.5)
0.5(1 )9.5I(0,1)() .
,
(0, U ]. Mathematica
Ppi( 6 0.1708|x = 0) 0.95, 95% (0, 0.1708] ( 3.2).
x = 3.
Beta(3.5, 7.5)
(|x = 3) = (11)(3.5)(7.5)
2.5(1 )6.5I(0,1)() .
, .
[L, U ] L U
UL
(|x = 3)d = 1
(L|x = 3) = (U |x = 3) .
-
62 3.
015 8.8277 0.8231
Ppi(8.8277 6 6 0.8231|x = 0) = 0.95
3.3: 95% =logit() 3.4.4.
Mathematica 1 = 0.95 L 0.0745 U 0.5795, 95% [0.0745, 0.5795]
( 3.2).
, x = 10.
Beta(10.5, 0.5)
(|x = 10) = (11)(10.5)(0.5)
9.5(1 )0.5I(0,1)() .
,
[L, 1).
( 0.5) x = 0, Ppi( > 10.1708|x =10) = 0.95, 95%
[0.8292, 1), 1 0.1708 = 0.8292 ( 3.2).
-
. = g() -
[L, U ], [L, U ] 100(1 )% , , [L, U ] 6= [g(L), g(U )]. : 3.31.
3.4.4. 3.4.3 -
= g() = logit() := log[/(1 )]. x = 0.
(|x = 0) = (11)(0.5)(10.5)
e0.5
(1 + e)11IR() .
( 3.16.) 3.3 95% -
( Mathematica), [8.8277,0.8231]. 3.4.3, (0, 0.1708].
logit(0) := lim0 log[/(1 )] = , logit(0.1708) = 1.5800 (,1.5800] 6=[8.8277,0.8231].
-
3.4. 63
-
.
() . ,
-
.
.
3.4.3. [ ]
. 100(1)% [1/2 , /2].
. ()
Ppi( > |x
) =
(|x)d = .
(
3.11.)
Ppi( < 1/2|x
) = Ppi( > /2|x
) = /2.
-
.
3.4.5. 3.4.3. -
x = 0 (|x = 0) Beta(0.5, 10.5)
Ppi( < 0.00004789|x = 0) = Ppi( > 0.2172|x = 0) = 0.025,
[0.00004789, 0.2172] 95% . (
(0, 0.1708].) 95% -
= logit() [logit(0.0000479), logit(0.2172)] =
[9.9466,1.2821]. x = 3 (|x = 3) Beta(3.5, 7.5)
Ppi( < 0.0927|x = 3) = Ppi( > 0.6058|x = 3) = 0.025,
[0.0927, 0.6058] 95% . (
[0.0745, 0.5795].) 95%
logit() [2.2811, 0.4297].
-
64 3.
, x = 10 (|x = 10) Beta(10.5, 0.5)
Ppi( < 0.7828|x = 10) = Ppi( > 0.999952|x = 0) = 0.025,
[0.72828, 0.999952] 95% . (
[0.8292, 1).) 95%
logit() [1.2821, 9.9466].
:
logit() x = 0 x = 10;
3.5
3.5.1
, H0 :
0 H1 : 1 d D = {0, 1}, 1 H0 0 .
01
L(d, ) =
{d, 0 ,1 d, / 0 .
D = {0, 1}
(|x) =
L(, )(|x
)d
=
0
(|x)d +
c
0
(1 )(|x)d
= Ppi( 0|x) + (1 )Ppi( c0|x
)
= Ppi( c0|x) + {Ppi( 0|x
) Ppi( c0|x
)}.
Ppi( 0|x) > Ppi( c0|x
), Ppi( 0|x
) > 1/2, (|x
)
= 0,
= 1. Bayes
pi(x) =
{1, Ppi( c0|x
) > Ppi( 0|x
) ,
0, =
{1, Ppi( 0|x
) < 1/2,
0, Ppi( 0|x) > 1/2,
Bayes H0
50% .
( 3.17)
01 (3.1.5), Bayes
pi(x) =
{1, Ppi( 0|x
) < 1/(0 + 1) ,
0, Ppi( 0|x) > 1/(0 + 1) .
(3.5.16)
-
3.5. 65
3.5.2 Bayes
p0 p1 = 1 p0 H0 : 0 H1 : 1. a0(x
) a1(x
) = 1 a0(x
)
( x). 2.2.1, p1/p0
a1(x)/a0(x
)
H1 H0.
3.5.1. [ Bayes (Bayes factor) ]
H1 H0
H1 H0,
B10(x) =
a1(x)/a0(x
)
p1/p0=
a1(x)
a0(x) p0p1, (3.5.17)
Bayes H1.
Bayes H0:
H0 H1 H0
H1,
B01(x) =
a0(x)/a1(x
)
p0/p1=
1
B10(x).
Bayes H1
H1 H0 .
Jeffreys :
Jeffreys (1961)log10B10(x
) B10(x
) H0
0 0.5 1 3.2 0.5 1 3.2 10 1 2 10 100 > 2 > 100
(, log10 x .) -
. Kass and Raftery (1995)
( ) , .
Kass and Raftery (1995)2 logB10(x
) B10(x
) H0
0 2 1 3 2 6 3 20 6 10 20 150 > 10 > 150
(, log x .)
-
66 3.
3.5.1. 2.3.4.
: IQ 100 (
) 100 (
). H0 : 6 100
H1 : > 100. N (100, 225), ( 100
). x = 115,
N (110.39, 69.23). H0
a0(x) = Ppi( 6 100|x) = P
(Z 6
100 110.3969.23
)= (1.245) 0.106
( Z N (0, 1)) H1 a1(x) = 1 a0(x) = 0.894. , Bayes H1
B10(x) =
a1(x)/a0(x)
p1/p0=
0.894/0.106
0.5/0.5= 8.43,
,
.
Bayes .
0 = {0} 1 = {1} ( )
B10(x) =
(1|x)
(0|x) (0)(1)
=f(x
|1)(1)/m(x
)
f(x|0)(0)/m(x
) (0)(1)
=f(x
|1)
f(x|0) ,
Bayes -
.
, H0 : 0 H1 : 1 : .
.
H0 0() 0
H1 1() 1. p0 p1 = 1 p0 ,
() = p00()I0() + p11()I1() =
{p00() , 0 ,p11() , 1 .
(3.5.18)
, 1 = c0, p0 = P
pi( 0), p1 = Ppi( 1) = 1 p0, 0() = p10 ()I0(), 1() = p11 ()I1(). ( 3.5.1.)
, .
, H0
0(|x) =
f(x|)0()I0()
0f(x
|)0()d =
f(x|)0()m0(x
)
I0() ,
-
3.5. 67
m0(x) =
0
f(x|)0()d
H0, H1
1(|x) =
f(x|)1()I1()
1f(x
|)1()d =
f(x|)1()m1(x
)
I1() ,
m1(x) =
1
f(x|)1()d
H1. ,
(|x) =
f(x|)()
f(x|)()d =
p0f(x|)0()
m(x)
I0() +p1f(x
|)1()
m(x)
I1() ,
,
m(x) =
f(x
|)()d
= p0
0
f(x|)0()d + p1
1
f(x|)1()d
= p0m0(x) + p1m1(x
) .
,
a0(x) =
0
(|x)d =
p00
0()f(x|)d
m(x)
=p0m0(x
)
p0m0(x) + p1m1(x
)
(3.5.19)
a1(x) =
1
(|x)d =
p11
1()f(x|)d
m(x)
=p1m1(x
)
p0m0(x) + p1m1(x
)= 1 a0(x
) .
(3.5.20)
3.5.1. Bayes H1
H1
H0, ,
B10(x) =
m1(x)
m0(x).
. (3.5.17), (3.5.19) (3.5.20)
B10(x) =
a1(x)
a0(x) p0p1
=p1m1(x
)/m(x
)
p0m0(x)/m(x
) p0p1
=m1(x
)
m0(x).
3.5.1. Bayes
() . ()
.
Bayes p0 p1
, .
-
68 3.
3.5.2. ( 3.18) -
Bayes
a0(x) =
{1 +
1 p0p0
B10(x)
}1.
, 01,
Bayes H1
a0(x)
p00p11
[ (3.5.16)]. , a0(x)
( ) .
p ,
.
3.5.3. H0 : = 0 ().
H0 () Dirac 0,
0. ,
m0(x) =
0
f(x|)0()d = f(x
|0)
Bayes H1 B10(x) = m1(x
)/f(x
|0).
3.5.2. X N (, 1), = R, H0 : = 0 H1 : 6= 0. H1 N (, 2), = 0. , x ,
m0(x) = f(x| = 0) = 12
ex2/2
m1(x) =
R
{12
e(x)2/2
}{
1
2
e2/22
}d =
12(1 + 2)
ex2/[2(1+2)] .
H0
a0(x) =
{1 +
1 p0p0
11 + 2
exp
(x22
2(1 + 2)
)}1.
3.2 H0 : = 0
Bayes H1 x, p0 = 1/2 ( -
) = 1. x = 2.58,
H0
(p = 0.001), .
: 3.19.
-
3.5. 69
x 0 1.28 1.64 1.96 2.58
a0(x) 0.5858 0.4842 0.4193 0.3512 0.2112B10(x) 0.71 1.07 1.39 1.85 3.73
3.2: H0 Bayes H1 x 3.5.2.
3.5.3. [Kass and Raftery (1995) ]
.
.
X1/n1, . . . ,X/n :
i Xi ni . H0 : Xi B(ni, ), i =1, . . . , , (
) H1 : Xi B(ni, i) : .
3.3.5
= 5 . H0 Beta(14, 6). H1 1, . . . , 5 Beta(14, 6).,
m0(x) =
10
({5i=1
(nixi
)I{0,1,...,ni}
}
xi(1 )
ni
xi)(
(20)
(14)(6)141(1 )61
)d
={5
i=1
(nixi
)I{0,1,...,ni}
} (20)(14)(6)
10481(1 )251d
={5
i=1
(nixi
)I{0,1,...,ni}
} (20)(14)(6)
(48)(25)
(73),
logm0(x) = log
i=1
(nixi
)+ log
{(20)(48)(25)
(14)(6)(73)
}= log
i=1
(nixi
) 35.39.
,
m1(x) =
5i=1
10
({(nixi
)I{0,1,...,ni}
}xii (1 i)nixi
)
((20)
(14)(6)141i (1 i)61
)di
={5
i=1
(nixi
)I{0,1,...,ni}
} ( (20)(14)(6)
)5 5i=1
1014+xi1(1 )6+nixi1d
={5
i=1
(nixi
)I{0,1,...,ni}
} ( (20)(14)(6)
)5 5i=1
(14 + xi)(6 + ni xi)(20 + ni)
,
-
70 3.
logm1(x) = log
i=1
(nixi
)+ 5 log
{(20)
(14)(6)
}+
5i=1
log
{(14 + xi)(6 + ni xi)
(20 + ni)
}
= logi=1
(nixi
) 35.18.
Bayes H1 B10(x) = m1(x
)/m0(x
)
2 logB10(x) = 2{logm1(x
) logm0(x
)} = 2 (35.18 + 35.39) = 0.42.
H1
5 .
: 3.20.
3.6
. ,
x f(x
|)
Y g(y|). Y X
.
3.6.1. [ (Predictive distribution) ]
() Y
x
(y|x) =
g(y|)(|x
)d . (3.6.21)
. X1, . . . ,Xn, Y
, Y,X,
g(y|)f(x|)() .
Y,X
,
Y,Xg(y|)f(x
|)()d =
g(y|)(|x
)m(x
)d = m(x
)
g(y|)(|x
)d,
m(x) =
f(x
|)()d X
x
. ,
m(x), Y X
= x
,
(y|x) (3.6.21).
-
3.6. 71
3.6.1. X1, . . . ,Xn
N (, 2), = R > 0 . () N (, 2) x
,
(|x) N
(n2x+ 2
n2 + 2,
22
n2 + 2
).
(x) 2(x
) -
. , Y N (, 2) , x
3.6.1
(y|x) =
(1
2
exp
{(y )
2
22
})(
1
(x)2
exp
{( (x))
2
22(x)
})d
=1
2 (x)exp
{12
(y2
2+(x
)2
2(x)
)}
exp
{12
[(1
2+
1
2(x)
)2 2
(y
2+
(x)
2(x)
)
]}d
=1
2 (x)exp
12
(y2
2+(x
)2
2(x)
)(
1
2+
1
2(x)
)1( y2
+(x
)
2(x)
)2
exp
{12
(1
2+
1
2(x)
)[2 2
(1
2+
1
2(x)
)1( y2
+(x
)
2(x)
)
+
(1
2+
1
2(x)
)2( y2
+(x
)
2(x)
)2 d
=1
2 (x)exp
12
(y2
2+(x
)2
2(x)
)(
1
2+
1
2(x)
)1( y2
+(x
)
2(x)
)2
exp
12
(1
2+
1
2(x)
)[
(1
2+
1
2(x)
)1( y2
+(x
)
2(x)
)]2 d.
2
(1
2+
1
2(x)
)1/2 -
N((
1
2+
1
2(x)
)1( y2
+(x
)
2(x)
),
(1
2+
1
2(x)
)1)
R .
(y|x) =
122 + 2(x
)exp
{ (y (x))
2
2(2 + 2(x))
}
-
72 3.
N ((x), 2 + 2(x
)).
Y . , 100(1 )%, [
(x) z/2
2 + 2(x
) , (x
) + z/2
2 + 2(x
)].
: 3.22, 3.23, 3.24.
3.7
.
, -
(
).
N (, 2) ( ) .8
2 .
() N (0, 2)
(|x) N
(n2x+ 20n2 + 2
,22
n2 + 2
).
3.7.1
X
= (X1, . . . ,Xn) N (, 2), = = (0,), R .
f(x|2) = 1
(2)n/2(2)n/2exp
{ 122
(xi )2
}IRn(x
) .
f(x|2) 2
( ). ,
2 IG(, ),
(2|x) f(x
|2)(2)
=
(1
(2)n/2(2)n/2exp
{ 122
(xi )2
})(
e/2
()(2)+1I(0,)(
2)
)
8 2 .
-
3.7. 73
1(2)+
n2+1
exp
{ 12[ + 12
(xi )2
]}I(0,)(
2) ,
2 ,
IG (+ n2 , + 12 (xi )2). + n/2 > 1, 2 [ Bayes L(d, 2) = (d 2)2]
pi(x) =
+ 12
(xi )2+ n2 1
=2 +
(xi )2
2( 1) + n . (3.7.22)
> 1,
pi(x) =
n
2( 1) + n
(xi )2n
+2( 1)
2( 1) + n
1 ,
(xi )2/n ( 2 )
/( 1).
L(d, 2) =1
2(d 2)2 . (3.7.23)
Bayes ( 3.4)
pi(x) =
Epi{(1/2)2|x}
Epi{1/2|x} =
1
Epi{1/2|x} .
2 1/2
, Epi{1/2|x} = (+n/2)/(+
(xi )2/2), Bayes
pi(x) =
2 +
(xi )22+ n
.
Bayes
Bayes (3.7.22).
() = 1/,
(
X1 , . . . ,Xn ).
(|x) f(x
|)()
=
(1
n(2)n/2exp
{ 122
(xi )2
}) 1I(0,)()
1n+1
exp
{ 122
(xi )2
}I(0,)() ,
-
74 3.
2 ( )
(2|x) 1
(2)n2+1
exp
{ 122
(xi )2
}I(0,)() ,
IG(n/2,(xi )2/2), (xi )2/(n 2). ( )
Bayes .
(3.7.23) Bayes
(xi )2/n. : 3.25.
3.7.2
X
= (X1, . . . ,Xn) N (, 2), = (, ) = R (0,).
f(x|) = f(x
|, ) = 1
n(2)n/2exp
{ 122
(xi )2
}IRn(x
)
=1
n(2)n/2exp
{ 122
[n(x )2 + (n 1)s2]} IRn(x
) ,
s2 =
(xi x)2/(n 1). ni=1
(xi )2 =ni=1
(xi x)2 + n(x )2 .
2.3
.
,
, 2. ( )
2 2,
(, 2) = (|2)(2) .
(|2) N (0, 2/n0), 2 IG(, ), 0 R, 0, , > 0 , .
2 2 n0
n.
= (, 2)
(, 2|x) f(x
|, 2)(|2)(2)
(
1
(2)n/2exp
{ 122
[n(x )2 + (n 1)s2]}) (3.7.24)
-
3.7. 75
(1
(2)1/2exp
{ n022
( 0)2} 1
(2)+1exp
{ 2
})
(1
exp
{n+ n0
22
( nx+ n00
n+ n0
)2}) (3.7.25)
(1
(2)n/2++1exp
{ 12
( +
nn0(x 0)22(n + n0)
+(n 1)s2
2
)}).
( 3.26.)
(, 2|x) = (|2, x
)(2|x
)
(3.7.25)
(|2, x) N
(nx+ n00n+ n0
,2
n+ n0
),
(2|x) IG
(+
n
2, +
nn0(x 0)22(n + n0)
+(n 1)s2
2
).
.
2
( 3.26).
, 2
.
(nx + n00)/(n + n0), 2
.
,
(|x).
2:
(|x) =
2=0
(, 2|x)d2
2=0
1
(2)+(n+1)/2+1exp
{ 12
[ +
nn0(x 0)22(n+ n0)
+(n 1)s2
2+n+ n0
2
( nx+ n00
n+ n0
)2]}d2
[ +
nn0(x 0)22(n + n0)
+(n 1)s2
2+n+ n0
2
( nx+ n00
n+ n0
)2](+(n+1)/2)
[1 +
n+ n0
2 + nn0(x0)2
n+n0+ (n 1)s2
( nx+ n00
n+ n0
)2](n+2+1)/2
Tn+2nx+ n00
n+ n0,2 + nn0(x0)
2
n+n0+ (n 1)s2
(n+ n0)(n+ 2)
-
76 3.
( T ). > 0, n + 2 > 1 -
Bayes . tm,
Tm(0, 1) ( T m ), 100(1 )%
nx+ n00n+ n0
tn+2,
2 + nn0(x0)2n+n0 + (n 1)s2(n+ n0)(n + 2)
.
3.7.1. -
. 2
.
. . ,
-
.
. 2
( ).
.
, .. N (, 2) 2 IG(, ) ( 3.27). .
= (, ) (, ) = 1/,
.
= 2, (, ) Jeffreys (
2.4.11),
= 1, ,
( ) () = 1, () = 1/,
= 0, (, ) R (0,).
, (, ) .
= (, )
(, |x) f(x
|, )(, )
(
1
nexp
{ 122
[n(x )2 + (n 1)s2]}) 1
=
(1
exp
{ n22
( x)2})
(
1
n+1exp
{(n 1)s
2
22
}),
-
3.8. 77
, 2,
(, 2|x)
(1
exp
{ n22
( x)2})
(
1
(2)n+2
2+1
exp
{(n 1)s
2
22
})
(|2, x)(2|x
) , (3.7.26)
(|2, x) N
(x,2
n
),
(2|x) IG
(n+ 2
2,(n 1)s2
2
).
,
(3.7.26) 2. ( )
(|x) =
0
(, 2|x)d2
[1 +
n( x)2(n 1)s2
]n+12
Tn+2(x,
(n 1)s2n(n+ 2)
).
T , n + 2 > 0. Jeffreys = 2,
n = 1.
,
.
(, ) = 1/, = 1. n > 2
T . . = 1 , Tn1(x, s2/n). ,
( ) xs/n Tn1(0, 1) ,
x, s
. , 100(1 )% ( )
[x tn1,/2
sn, x+ tn1,/2
sn
].
100(1)% . , -
2 IG(n12 ,
(n1)s2
2
), (n 1)s2/2 G((n 1)/2, 1/2) 2n1,
. 100(1 )% 100(1 )% . (
.)
-
78 3.
3.8
Y. ,
( ) . , a Rp,
a =
a1...
ap
(a1, . . . , ap) .
, A p q , A , qp A.
3.8.1
Y = (Y1, . . . , Yp) p
= (1, . . . , p) Rp 9 = (ij)
f(y) =1
(2)p/2||1/2 exp{12(y )1(y )
}IRp(y) . (3.8.27)
Y Np(,). :
q < p Y1, . . . , Yp q .
, : Yi N (i, ii), i = 1, . . . , p.
, i E(Yi) = i, Var(Yi) = ii. ,
i, j Cov(Yi, Yj) = ij .
, = diag(11, . . . , pp), -
Y1, . . . , Yp .
Y1, . . . , Yp
N (, 2) Y = (Y1, . . . , Yp) - = (, . . . , )
= 2Ip, Ip p.
Y Np(,) A q p , b Rq, AY + b Nq(A +b,AA).
, a Rp aY N (a,aa) ( ).
y
(3.8.27)
f(y) exp{12
(y1y y1 1y+ 1)} IRp(y)
9 p p A x Rp, x 6= 0 := (0, . . . , 0) xAx > 0.
-
3.8. 79
exp{12
(y1y 2y1)} IRp(y) (3.8.28)
[ y1 = (y1) = 1y]. , -
Np(,) (3.8.28).
. ( , ,
-
.) Y = (Y1, . . . ,Yn) Np(,), = = Rp.
f(y|) =ni=1
f1(yi|) =ni=1
1
(2)p/2||1/2 exp{12(yi )1(yi )
}IRp(yi)
=1
(2)np/2||n/2 exp{12
ni=1
(yi )1(yi )}
IRpn(y) , Rp .
f(y|) (= ),
f(y|) exp{12
ni=1
(yi
1yi yi1 1yi + 1)}
IRp()
exp{12
ni=1
(21yi + 1)}
IRp()
= exp
{12
(n1 2n1y)} IRp() ,
y :=yi/n. ( (3.8.28),
Np(y, n1).)
() Np(,T), Rp T pp .
() exp{12
(T1 2T1)} IRp()
(|y) f(y|)() exp
{12
(n1 2n1y)}exp
{12
(T1 2T1)} IRp()
exp{12
[(n1 +T1) 2(n1y+T1)]} IRp()
= exp
{ 1
2
[(n1 +T1)
-
80 3.
2(n1 +T1)(n1 +T1)1(n1y +T1)]}
IRp() .
(3.8.28), p
Epi{|y} = (n1 +T1)1(n1y +T1)
Cov{|y} = (n1 +T1)1 .
, Bayes
pi(y) = (n1 + T1)1(n1y + T1).
,
. .. 1,
(1, 0, . . . , 0) = 1, Bayes (1, 0, . . . , 0)pi(y)
pi(y).
1 2, (1,1, 0, . . . , 0) = 1 2, Bayes (1,1, 0, . . . , 0)pi(y) pi(y).
() = 1
( 3.28).
(|y) f(y|) exp{12
(n1 2n1y)} IRp() ,
Np(y, n1). ( ,
.)
3.8.2
Y = (Y1, . . . , Yp) p -
m N,
= (1, . . . , p) := {0 < i < 1, i = 1, . . . , p,
pi=1 i < 1, } (3.8.29)
f(y) =m!
y1! . . . yp!(m y1 . . . yp)! y11 . . .
ypp (1 1 . . . p)my1...yp
=m!
(p
i=1 yi!) (mp
i=1 yi)!(p
i=1 yii ) (1
pi=1 i)
mp
i=1 yi ,
y1, . . . , yp {0, 1, . . . ,m},p
i=1
yi 6 m.
-
3.8. 81
Y Mp(m; 1, . . . , p).
. : p + 1 -
. i i, i = 1, . . . , p,
(p+1) 1pi=1 i. m . Y1, . . . , Yp p (p+1)
mpi=1 Yi. :
q < p Y1, . . . , Yp .
, : Yi B(m, i), i = 1, . . . , p.
, i E(Yi) = mi, Var(Yi) = mi(1 i)., i 6= j Cov(Yi, Yj) = mij.
Y = (Y1, . . . ,Yn) Mp(m; 1, . . . , p), (3.8.29). Yi = (Yi1, . . . , Yip)
,
( )
f(y|) =ni=1
f1(yi|) ni=1
(pj=1
yijj
)(1pj=1 j)m
pj=1 yij
=(p
j=1 n
i=1 yijj
)(1pj=1 j)nm
ni=1
pj=1 yij
, .
( (n
i=1 Yi1, . . . ,n
i=1 Yip) -
.)
= (1, . . . , p). -
p Dirichlet. X = (X1, . . . ,Xp)
p Dirichlet 1, . . . , p, p+1 > 0
f(x) =(1 + . . . + p + p+1)
(1) . . .(p)(p+1)x111 . . . x
p1p (1 x1 . . . xp)p+11
=(p+1
i=1 i
)p+1
i=1 (i)
(pi=1 x
i1i
)(1pi=1 xi)p+11 , x1, . . . , xp (0, 1) , pi=1 xi < 1.
x Dp(1, . . . , p;p+1). Dirichlet .
:
q < p X1, . . . ,Xp Dirichlet.
, : Xi Beta(i,j 6=i j), i = 1, . . . , p.
-
82 3.
, i E(Xi) = i/p+1
j=1 j, Var(Xi) =
i
j 6=i j/[(
j)2 (1 +
j)].
, i 6= j Cov(Xi,Xj) = ij/[(
j)2 (1 +
j)].
() Dp(1, . . . , p;p+1).
(|y) f(y|)()
{(pj=1
ni=1 yij
j
)(1pj=1 j)nm
ni=1
pj=1 yij
}{(p
j=1 j1j
)(1pj=1 j)p+11
}I()
=(p
j=1 j+
ni=1 yij
j 1)(
1pj=1 j)p+1+nmn
i=1
pj=1 yij1
I() ,
DirichletDp(1+
yi1, . . . , p+
yip;p+1+nm
yij).
Epi{|y} = Epi{(1, . . . , p)|y} =
(1 +
yi1
nm+
j, . . . ,
p +
yi1nm+
j
),
Bayes .
j
.
3.9
3.1. g() g() = [0,). .
() L(d, ) =d
g() log d
g() 1
() L(d, ) = {log d log g()}2
() L(d, ) =d
g()+g()
d 2
.
d = g() . -
d/g() t = d/g()
t > 0.
3.2. (, ) -
(, )
w() > 0.
3.3.
.
-
3.9. 83
: g() A g. c A T = T (X
) = c (
g() c) g().
. () c A, 0 g(0) = c.() : Y E(Y ) = 0 Y = 0
.
3.4. L(d, ) =
w(){d g()}2, w() > 0. Bayes
piw(x) :=
Epi{w()g()|x}
Epi{w()|x} .
3.5. X
= (X1, . . . ,Xn) N (, 2), =R, > 0 () = 1, R.() Bayes g() = 2
L(d, ) = {d 2}2, pi = pi(X) = X2 + 2/n.
() Bayes
2.
Bayes .
() ;
( ;) ;
3.6. X
= (X1, . . . ,Xn) Bernoulli B(1, ), =(0, 1), L(d, ) =
(d )2(1 ) . ( ) Bayes
() () Beta(, ), , > 0() () = [(1 )]1.
3.7. 3.3.5:
() -
Jeffreys, Beta(1/2, 1/2).() () L(d, ) =
(d )2(1 ) .
3.8. X
= (X1, . . . ,Xn) Poisson P(), =(0,). () G(, ), , > 0, , Bayes g() = P(X1 = 0)
pi = pi(X) =
( + n
1 + + n
)+Xi.
3.9. X
= (X1, . . . ,Xn) Poisson P(), =(0,), L(d, ) = 1
(d )2. ( ) Bayes
-
84 3.
() () G(, ), , > 0,() () Jeffreys.
3.10. 3.3.7:
() L(d, ) =1
(d )2
() G(1.8, 0.9).() Xi P(iti), i = 1, . . . , 10 ( i), 1, . . . , 10 G(1.8, 0.9), i
().
3.11. () 3.3.2 d > m.
()
L(d, ) =
{(1 )(g() d) , d 6 g()(d g()) , d > g() ,
0 < < 1 . -
d
g().
. q
P(Y 6 q) > 1 P(Y > q) >
, ,
P(Y < q) 6 1 P(Y > q) 6 Y .
3.12. a 6= 0,
La(d, ) = ea(dg()) a(d g()) 1
g()
Bayes g()
pi(x) = 1
aEpi{eag()
x
}.
3.13. 3.1
Bayes g().
3.14. () Beta(, ), , > 0.() X B(n, ), n > 2, Beta(1/2, 1/2), 3.3.9.
() n > 3, () [(1 )]1I[0,1]().
-
3.9. 85
3.15. X
= (X1, . . . ,Xn) Poisson P(), =[0,), () G(, b), , > 0. - .
3.16. Beta(, ), , > 0,
:= logit() (+ )()()
e
(1 + )+IR() .
, 0 := log(/).
[L, U ] < L < U 1/(0 + 1) .
3.18. H0
a0(x) =
{1 +
1 p0p0
m1(x)
m0(x)
}1.
3.19. X
= (X1, . . . ,Xn) E(), =(0,) () G(, ), , > 0 .() H0 : = 0 H1 : 6= 0 Bayes H1.
() 0 = 2, = 1, = 1, x
= (3.55, 1.95, 1.10, 1.88, 2.75),
;
3.20. 3.3.7 3.10:
H0 : Xi P(ti), i = 1, . . . , 10, H1 : Xi P(iti). H0 G(1.8, 0.9) H1 1, . . . , 10 G(1.8, 0.9), 2 logB10(x
) = 84.84. ;
3.21. N (, 16), = R,
10i=1 xi = 100. N (8, 25).
() N (, 16).() 95% .
3.22. X
= (X1, . . . ,Xn) Poisson P(), = (0,), () G(, ), , > 0. x
Y P()
, Y x
NB(+ nx,
+ n
+ n+ 1
).
-
86 3.
3.23. X
= (X1, . . . ,Xn) Bernoulli B(1, ), =(0, 1), () Beta(, ), , > 0. x
Y B(1, ) , Y x
B
(1,
+ nx
+ + n
).
3.24. X E(), = (0,) (E(X) = 1/), () G(, ), , > 0. x X Y E() , Y x Pareto Par(II)(+ 1, + x).. Pareto Par(II)(, ), , > 0,
f(t) =
( + t)+1I(0,)(t) .
3.25. X
= (X1, . . . ,Xn) N (, 2), = = (0,), R .() L(d, 2) = (d 2)2, - ( ) 2 c
(Xi )2
c = 1/(n + 2).
() ( )
Bayes;
() () -
(3.7.23).
3.26. () (3.7.24) (3.7.25).
() 2 L(d, ) = {d 2}2/2.
3.27. X
= (X1, . . . ,Xn) N (, 2), = (, 2) = R (0,). 2 N (, 2), 2 IG(, ), R, 2, , > 0 .() , 2 .
() 2 .
2
2;
() 2;
; ;
3.28. Y Np(,), = = Rp. Jeffeys () = 1.
.
3.29.* [DeGroot (1970) ] .
( )
-
3.9. 87
14 34 .
W 12
P(W = ) = = 1 P(W = ) .
W = W = .
01, 12 < 0, 0 < p < 1. : N, X Bernoulli
p.
:
f(x) =( + x)
()
p(1 p)xx!
IZ+(x) .
y (y) = (y1)!, ,
f(x) =
( + x 1
x
)p (1 p)xIZ+(x) .
: E(X) = (1 p)/p, Var(X) = (1 p)/p2.
89
-
90 .
: MX(t) = E[exp(tX)] =
[p
1 (1 p)et]
, t < log(1 p). :
X1 NB(1, p), . . . ,Xm NB(m, p) .. ( p),
mi=1Xi NB(
mi=1 i, p).
= 1 .
N := {1, 2, . . .}, {, + 1, . . .}. Y N p (0, 1) Bernoulli p .
Bernoulli
. , X NB(, p), Y = X + . ( Z+)
(0,) ( ).
(Geometric)
: X Ge(p), 0 < p < 1. :
Bernoulli
p.
:
f(x) = p(1 p)xIZ+(x) . : E(X) = (1 p)/p, Var(X) = (1 p)/p2. : MX(t) = E[exp(tX)] =
p
1 (1 p)et t < log(1 p).
Poisson
: X P(), > 0. :
f(x) = ex
x!IZ+(x) .
: E(X) = , Var(X) = .
: MX(t) = E[exp(tX)] = exp{(et 1)}, t R. :
X1 P(1), . . . ,Xm P(m) Poisson .., m
i=1Xi P(mi=1 i).
-
91
.2
(Uniform)
: X U(, ), < R. :
f(x) =1
I(a,b)(x) .
: E(X) = (+ )/2, Var(X) = ( )2/12. : MX(t) = E[exp(tX)] =
et ett( ) , t R\{0}, MX(0) = 1.
(Normal)
: X N (, 2), R, > 0. :
f(x) =1
2
e1
22(x)2
IR(x) .
: E(X) = , Var(X) = 2.
: MX(t) = E[exp(tX)] = et+2t2/2, t R.
:
X1 N (1, 21), . . . ,Xm N (m, 2m) .. a1, . . . , am R ,
mi=1 aiXi N (
mi=1 aii,
mi=1 a
2i
2i ).
= 0, = 1 . -
(
F ).
X N (, 2) Z = (X )/ N(0, 1) ( ). N (0, 1), z P(Z > z) = , Z N (0, 1), z.
(Gamma)
: X G(, ), , > 0. :
f(x) =x1ex
()I(0,)(x) .
: E(X) = /, Var(X) = /2.
: MX(t) = E[exp(tX)] = (1 t/), t < . :
X1 G(1, ), . . . ,Xm G(m, ) .. ( ),
mi=1Xi G(
mi=1 i, ).
= 1 () .
= m/2, m , = 1/2
m 2m.
-
92 .
(Beta)
: X Beta(, ), , > 0. :
f(x) =1
B(, )x1(1 x)1I(0,1)(x) .
: E(X) = /( + ), Var(X) = /[( + )2(+ + 1)].
:
X Beta(, ) 1X Beta(, ). X1 G(1, ),X2 G(2, ) .. ( ), X1/(X1 +X2) Beta(1, 2).
= = 1 U(0, 1).
(Inverse Gamma)
: X IG(, ), , > 0. :
f(x) =
()
1
x+1e/xI(0,)(x) .
: E(X) = /( 1) > 1, Var(X) = 2/[( 2)( 1)2] > 2.
:
X IG(, ) 1/X G(, ).
(Exponential)
: X E(), > 0. :
f(x) = exI(0,)(x) .
: E(X) = 1/, Var(X) = 1/2.
: MX(t) = E[exp(tX)] = (1 t/)1, t < . :
X1, . . . ,Xm .. E(), m
i=1Xi G(m,).
-
93
(Chi squared)
: X 2m, m N. : Z1, . . . , Zm ( )
N (0, 1), mi=1
Z2i 2m ,
m .
:
f(x) =xm21ex/2
(m2
)2m2
I(0,)(x) .
: E(X) = m, Var(X) = 2m.
: MX(t) = E[exp(tX)] = (1 2t)m/2, t < 1/2. :
2m G(m/2, 1/2). 2m, x P(X > x) = ,
X 2m, 2m,.
F
: X Fm1,m2 , m1,m2 > 0. : Y1 G(m1/2, 1/2), Y2 G(m2/2, 1/2)
X =Y1/m1Y2/m2
Fm1,m2 ,
F m1 m2 . :
f(x) =(m1+m2
2
)(m1/m2)
m1/2
(m12
)(m22
) xm1/21(1 +m1x/m2)(m1+m2)/2
I(0,)(x) .
: E(X) = m2/(m2 2) m2 > 2, Var(X) = 2m22(m1 +m2 2)/{m1(m2 2)2(m2 4)}, m2 > 4. :
X Fm1,m2 1/X Fm2,m1 . Fm1,m2, x P(X > x) =, X Fm1,m2, Fm1,m2,.
-
94 .
T
: X Tm(, 2), m > 0, R, > 0. : Z N (0, 1), Y G(m/2, 1/2) ,
X = ZY/m
+ Tm(, 2) .
:
f(x) =(m+12
)m
(m2
) (1 + (x )2m2
)m+12
IR(x) .
: E(X) = m > 1, Var(X) = m2/(m 2), m > 2. :
Tm(, 2) t m , . = 0, = 1 m N, Tm(0, 1) t m .
X Tm(, 2) (X )/ Tm(0, 1). m = 1 T1(, 2) Cauchy C(, ). (.)
X Tm(, 2) (X )2/2 F1,m. Tm(0, 1), x P(X > x) =, X Tm(0, 1), tm,.
m , Tm(0, 1) N (0, 1). Tm(, 2) N (, 2) .
Cauchy
: X C(, ), R, > 0. :
f(x) =1
1
1 + (x )2/2 IR(x) .
: , E(|X|) =. : ( t = 0).
: