Dealing with underdetermination in Bayesian models Bayesian models are supposedly well equipped to deal with high-dimensionality problems, and can handle sparse data well, too. But suppose I've created a model that estimate more parameters than there are data points. Are there tricks to deal with this?
 A: You say that there are several group-level parameters that you are interested in estimating. That sounds exactly like a hierarchical problem. Do some reading on hierarchical models and the key assumptions (like exchangability), and I think you'll find that your Bayesian background and this problem will easily fit into this paradigm. 
The classic example of hierarchical modeling is analyzing data on many students within several schools. We'd expect the process of students' learning to be similar generally, but that better schools would have students with higher achievement, that is, that there is correlation at the student level because each student is not entirely independent of other students in the same school. Accounting for the features of schools that make some better and others worse will help identify salient feature of education, and account for the systematic variation at the student level.
More generally, I would suggest fitting simpler versions of the model and then increasing complexity and the number of parameters gradually. It's much easier to wrap one's mind around, and you will probably learn interesting things about your data along the way which will help you to fit more complex, more realistic models.
Ultimately, what research design is sensible always depends on the research question, but from your description, it sounds like this problem is amenable to a hierarchical approach.
