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.

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() ,

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).

:

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