Past exam of the statistics course of the University of Oxford / 2021 / 3 / c

An unknown variance requires a joint prior distribution for $(\alpha ,\beta ,{\sigma }^2)$. The analysis is not determined until a prior for the unknown variance, and its relation to the coefficients, is specified. A maximum-likelihood estimate substituted for ${\sigma }^2$ does not by itself account for its posterior uncertainty.

For example, retain the independent coefficient prior distributions from part (b) and independently assign $s={\sigma }^2$ an inverse-gamma distribution with shape $a_0>0$ and scale $b_0>0$, whose probability density function is proportional to $s^{-a_0-1}{e}^{-b_0/s}$. Then the joint posterior density is proportional toConditional on $s$, the two coefficient posterior distributions are still the independent normal distributions in part (b), with $s$ substituted for ${\sigma }^2$. The other full conditional distribution isThe marginal coefficient posterior distribution requires marginalization over $s$; it is generally nonnormal and the two coefficients need not remain independent. In particular, the posterior cumulative distribution function of the requested mean becomesIts shortest 95% credible interval must be found from this marginal posterior distribution, for example by numerical integration or posterior simulation. This is normal regression with independent coefficient and variance priors.

A different, convenient choice is a normal-inverse-gamma prior, in which the conditional coefficient prior covariance matrix is proportional to ${\sigma }^2$. That choice gives a multivariate Student's t-distribution after marginalization, so the interval uses its univariate Student's t-distribution quantiles. It changes the coefficient prior specification; a Student interval does not follow automatically from retaining the fixed-${\tau }_0,{\tau }_1$ priors in part (b).