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Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2023

Paper 216

The <likelihood function> and <prior distribution> give the <posterior density>
$$
\pi(\beta\mid Y)\propto
\exp\left\{-\frac12(Y-X\beta)^T\Sigma_e^{-1}(Y-X\beta)
-\frac12\beta^T\Sigma^{-1}\beta\right\}.
$$
Completing the square in the <quadratic form> gives
$$
C=\left(X^T\Sigma_e^{-1}X+\Sigma^{-1}\right)^{-1},
\qquad
m=CX^T\Sigma_e^{-1}Y.
$$
Thus <Gaussian conjugacy for a normal linear model> yields
$$
\beta\mid Y\sim N(m,C).
$$

Solution

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

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

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

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