Exercise 13.4 - Marginal likelihood for linear regression

Answers

This is straightforward algebra:

\begin{aligned} p (𝒟 | γ) & = \int \int 𝒩 (𝐘 | 𝐗^{T} 𝐰, σ^{2} 𝐈) 𝒩 (𝐰 | 0, σ^{2} Σ) p (σ^{2}) d 𝐰 d σ^{2} \\ \propto \int \int {(σ^{2})}^{- \frac{N}{2} - \frac{D}{2} - 1} \exp {- \frac{1}{2 σ^{2}} [𝐰^{T} (𝐗^{T} 𝐗 + Σ^{- 1}) 𝐰 - 2 𝐰^{T} 𝐗 𝐘 + 𝐘^{T} 𝐘]} d 𝐰 d σ^{2} . \end{aligned}

We firstly integrate out $𝐰$ :

\int \exp {- \frac{1}{2 σ^{2}} [𝐰^{T} (𝐗^{T} 𝐗 + Σ^{- 1}) 𝐰 - 2 𝐰^{T} 𝐗 𝐘 + 𝐘^{T} 𝐘]} d 𝐰 \propto {[σ^{2} (𝐗^{T} 𝐗 + Σ^{- 1})]}^{- \frac{1}{2}} \exp {- \frac{S (γ)}{2 σ^{2}}} .

(It seems that two factors are lost in $S$ given by (13.17). Finally, we intergrate out $σ^{2}$ :

\int {(σ^{2})}^{- \frac{N + a}{2}} \exp {- \frac{S (γ)}{2 σ^{2}}} d σ^{2},

what left is tedious calculus. The plugging-in of the $g$ -prior is trivial.

solour_lfq

2021-03-24 13:42