݁௡~ࣨሺ0, ߪ௘^ଶሻൌࣨሺ0, ߙ^ିଵሻ

(3.50)

Gaussian distribution of errors is governed by a super distribution,

ߙ~ࣨሺ0, ߴሻ

(3.51)

rior of w is also designed as a Gaussian distribution [Bullen,

03], where ߚൌߪ௪^ିଶ,

ݓ௞~ࣨሺ0, ߚሻ

(3.52)

arameter ߚ is also controlled by a priori,

ߚ~ࣨሺ0, ߬ሻ

(3.53)

ose a set of all hyper-parameters (ߴ and ߬) is denoted by ߱, a

al probability (or likelihood) is denoted by ݌ሺܡ|ܟ, ߱ሻ, a

ed factor (evidence) is denoted by ݌ሺܡሻ, the a priori structure is

by ݌ሺܟ, ߱ሻ and a posterior probability is denoted by ݌ሺܟ, ߱|ܡሻ.

ionship between these notations is defined as below,

݌ሺܟ, ߱|ܡሻൌ^{݌ሺܡ|ܟ, ߱ሻ݌ሺܟ, ߱ሻ}

݌ሺܡሻ

(3.54)

priori structure ݌ሺܟ, ߴሻ can be decomposed as below,

݌ሺܟ, ߴሻൌ݌ሺܟ|ߴሻ݌ሺ߱ሻ

(3.55)

ܟ|߱ሻ is a conditional probability given hyper-parameters and

the a priori probability of hyper-parameters. With this

ation, the posterior probability is simplified as below, where the

term ݌ሺܡሻ is omitted because it is a constant,

ࣦ∝݌ሺܡ|ܟ, ߱ሻ݌ሺܟ|߱ሻ݌ሺ߱ሻ

(3.56)

bove three terms are defined as below one by one,