Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 36 / 3 / d / Solution

The regression coefficients have independent proper uniform distributions on very wide finite intervals. These are flat in the coefficients, not flat in fitted rates, and can be highly informative on the rate scale. The scale $\tau$ is uniform on $(0,100)$, and WinBUGS receives the corresponding precision $1/{\tau }^2$ for each normal distribution random effect. Writing $v={\tau }^2$, the induced prior density isIt is proper and integrable at zero.

The usual scale-invariant variance prior $p(v)\propto 1/v$ would instead cause an improper posterior from a log-uniform random-effect scale prior. After integrating out the plate effects, the likelihood function tends as $v\downarrow 0$ to the ordinary positive Poisson likelihood. On any compact interior set of coefficient values this limit is positive and bounded away from zero. Hence the posterior normalizing integral contains a divergent factor ${\int }_0^{\epsilon }dv/v$. There is no posterior distribution under that prior, even if formal full conditional distributions appear usable. Thus a proper prior with integrable mass near zero is needed here. The reciprocal-scale rule is the standard pure-scale Jeffreys prior; it should not be confused with the model-specific information prior for an additive variance component.