Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 216 / 1 / a

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- Solution a

Solution

The likelihood function and prior distribution give the posterior density

π (β ∣ Y) \propto exp {- \frac{1}{2} (Y - Xβ)^{T} Σ_{e}^{- 1} (Y - Xβ) - \frac{1}{2} β^{T} Σ^{- 1} β} .

Completing the square in the quadratic form gives

C = (X^{T} Σ_{e}^{- 1} X + Σ^{- 1})^{- 1}, m = C X^{T} Σ_{e}^{- 1} Y .

Thus Gaussian conjugacy for a normal linear model yields

β ∣ Y \sim N (m, C) .

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