I’m afraid I think I’m too new to Bayes to get my head around the exact question they are asking (or how you feel you messed up), but maybe if I describe why the correlation is of interest in my case, it might help?
My scenario is a survey of several potential ethical issues that might arise in a certain medical setting. Families indicated whether or not each issue was concerning to them, as a simple yes/no response.
The scientific questions of interest were, in order of complexity:
- What is the general level of concern about these ethical issues
- What is the general level of concern about each specific issue j
- How are the issues related to one another, e.g. are families who are likely to say ‘issue 7’ is a concern more or less likely to say ‘issue 3’ is a concern?
- [Still to do] What characteristics of families are associated with their general level of concern across all issues?
- [Still to do] Are there any family characteristics that are associated with concern about specific issues?
Using the notation from my previous post, \alpha_0 and \alpha_j help answer the first two questions; \gamma_i is a ‘random effect’ to allow families to have different levels of general concern across all issues, and \Omega is essentially there to answer the third question.
I had initially attempted to do this by putting a multivariate normal prior on the \alpha_j s and estimating their correlation, but I hope I was correct in realising (after many computational issues) that it was the correlation between issue-specific random effects within families (i.e. the \eta_{ij} s) that is actually what I wanted to model.
I agree with both of your posts in the other thread, but again I’m not sure if I’ve missed the point or not. I suppose that observing correlations in your posterior after prior independence is a different thing to obtaining a posterior for the correlation; and I hope my scenario is one in which the latter is of direct interest.