14.3Priors

Likelihood

Conjugate prior

Posterior hyperparameters

n

n

Bern (p)

Beta ( ; )

+

xi; + n

xi

i=1

n

i=1

n

n

X

X

Bin (p)

Beta ( ; )

+

xi; +

Ni xi

=1

n

i=1

i=1

Xi

X

X

NBin (p)

Beta ( ; )

+ rn; +

xi

n

=1

Xi

Po ( )

Gamma ( ; )

+

xi; + n

i=1

n

X

Multinomial(p)

Dir ( )

+

x(i)

=1

n

Xi

Geo (p)

Beta ( ; )

+ n; +

xi

=1

Xi

Likelihood

Conjugate prior

Posterior hyperparameters

m

(n)

n

Unif (0; )

Pareto(x

; k)

max

x

; xm

; k + n

Xi

xi

Exp ( )

Gamma ( ; )

+ n; +

02

P

=1

c2

N

N

n

02

c2

; 2

; 2

0

+

i=1 xi

=

1

+

n

,

1

n

c

1

0 0

+

02

c2

in=1(xi )2

N

c; 2

Scaled Inverse Chi-

+ n,

02 +

0

+ n

N

; 2

Normal-

+ nx

,

+ n,

+

n

,

Inverse

scaled

1

n

(x

)2

2

+ n

Gamma( ; ; ; )

Xi

x)2 +

+

(x

2(n + )

2

=1

i

MVN( ; c)

MVN( 0; 0)

1 + n 1

1

1 0

+ n 1x ,

0 1

+ n c 1 1

0

c

0

Wishart( ; )

n

T

MVN( c; )

Xi

Inverse-

n + ; + (xi c)(xi c)

=1

n

Xi

xi

Pareto(xmc ; k)

Gamma ( ; )

+ n; +

=1

log

xmc

x0; k0 kn

Pareto(xm; kc)

Pareto(x0; k0)

where

k0 > kn

n

Gamma ( c; )

Gamma ( 0; 0)

0 + n c; 0 +

Xi

xi

=1

If H0 : 2 0:

Prior probability P [H0] = Z 0

f( ) d

Posterior probability P [H0 j xn] = Z 0

f( j xn) d

Let H0; : : : ; HK 1 be K hypotheses. Suppose f( j Hk),

P

[H

k j

xn] =

f(xn j Hk)P [Hk]

;

K

n

[Hk]

Pk=1 f(x

j Hk)P

15