Last year (in fact I did some of this while travelling to go to my friend Lorenzo's stag do $-$ he's the one in white, but with no veil), I worked on a clinical paper discussing the prevalence of glaucoma with specific focus on the European population. The objective was relatively straightforward, except for the fact that the studies used to derive the estimations were quite heterogeneous and thus we could not pool them altogether.
So we used a nice (I think) Bayesian hierarchical model where different studies contributed to different parts of the estimation procedure. I built a model in which the overall prevalence was estimated using separate (but connected) modules $-$ basically age groups. So we first estimate a set of "level-1" parameters $\theta_1,\theta_2,\theta_3,\theta_4$ (effectively the age-group specific prevalences) using the observed data from the available studies. Some of these are assumed to be conditionally exchangeable, so that for example $\theta_2,\theta_3,\theta_4$ are used to inform the distribution of the parameter $\theta_5$, representing the prevalence among the over 50s. Again assuming conditionally exchangeability, $\theta_1$ and $\theta_5$ are used to inform the overall prevalence among the over 40s.
We have found sensible (or so I'm told by the clinicians!) results. It wasn't the place to brag about the use of a Bayesian approach, so the paper does not give much detail on the actual model. The observed data were counts of subjects with glaucoma in the study populations.
But it was cool that I persuaded them to report the results in graphical fashion and with the credible intervals.
Some of them do not even know that the model is Bayesian, but they were extremely happy with the results (or if they weren't, they were extremely nice to me anyway).
The paper is out now.