Ex. 3.7

Ex. 3.7

Assume yiN(β0+xiTβ,σ2), i=1,2,...,N, and the parameters βj are each distributed as N(0,τ2), independently of one another. Assuming σ2 and τ2 are known, show that the (minus) log-posterior density of β is proportional to i=1N(yiβ0jxijβj)2+λj=1pβj2 where λ=σ2/τ2.

Warning

The claim above does not seem to be correct.

Soln. 3.7

By Bayes' theorem we have

(1)P(β|y)=P(y|β)P(β)P(y).

By assumptions here we have

P(y|β)=1(2π)N/2σNexp(12σ2i=1N(yiβ0jxijβj)2),P(β)=1(2π)p/2σpexp(1τ2j=1pβj2).

Therefore, with λ=σ2/τ2, from (1) we have

ln(P(β|y))=12σ2(i=1N(yiβ0jxijβj)2+λj=1pβj2)+C,

where C is a constant independent of β.

The claim is true if and only if C=0, which is not the case here.