Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 36 / 4 / c

The bounded uniform prior distributions on the slopes and population intercept are proper and broad on their stated scale. They allow either sign but are not invariant to changes of units. The residual precision parameter $q=1/{\sigma }^2$ has a proper shape-rate $\mathrm{Gamma}(0.001,0.001)$ prior. Equivalently, ${\sigma }^2$ has inverse-gamma distribution with the same shape and scale. This is a commonly used diffuse choice, but it is not literally noninformative and its tail behaviour merits sensitivity analysis.

The prior for $\tau$ is uniform on $(0,100)$, so the induced density of $v={\tau }^2$ isIt is integrable at zero, unlike the usual scale-invariant variance kernel $\pi (v)\propto 1/v$. The latter kernel, often called the standard variance Jeffreys prior, would cause an improper posterior from a log-uniform random-effect scale prior here. After integrating the child intercepts, each child's response vector has a normal likelihood with covariance matrix ${\sigma }^2{I}_3+v11^T$. For fixed positive ${\sigma }^2$, this covariance remains nonsingular at $v=0$, and the marginal likelihood has a positive finite limit there. On a compact set of the other parameters its positivity givesConsequently the scale-invariant kernel cannot be normalized to a posterior distribution. A proper prior on the standard deviation avoids this problem. This familiar scale prior is not necessarily the actual Jeffreys prior for an additive variance component derived from the observed-data likelihood; the latter can be finite at zero.