= 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> is
$$
\boxed{p(v)=\frac1{200\sqrt v}\mathbf1_{(0,10000)}(v).}
$$
It is proper and integrable at zero.
The usual scale-invariant variance prior $p(v)\propto1/v$ 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 $v\downarrow0$ 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 $\int_0^\varepsilon dv/v$. There is no <posterior distribution> under that prior, even if formal full conditional distributions appear usable. Thus \b[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.
Back to article page
