Past exam of the statistics course of the University of Oxford / 2021 / 3 / b / i / Solution

Let $S_{xx}={\sum }_i{x}_i^2$, $S_{xy}={\sum }_i{x}_i{y}_i$, and $\bar{y}=n^{-1}{\sum }_i{y}_i$. The independent observations in the normal linear model giveThe centered design condition ${\sum }_i{x}_i=0$ removes the $\alpha \beta$ term:The stated prior density factors into independent normal distributions for the regression intercept $\alpha$ and regression coefficient $\beta$. Multiplication by the likelihood function therefore leaves separate quadratic expressions in the two coefficients. Define the positive posterior variancesand the posterior meansBy completing the square, the joint posterior density isEquivalently,This is Gaussian conjugacy for a normal linear model. Each posterior precision parameter is the sum of its data and prior precision parameters. The formulas remain valid when $S_{xx}=0$: all $x_i$ are then zero, so the posterior distribution of $\beta$ equals its prior distribution.