A Bayesian Subset Selection Model

Hello, I was wondering if anyone could comment on a Bayesian model for subset selection? The problem is thus:

Assume a plate contains 3 objects, A, B, and C. An individual can choose to select any subset of the items ranging from none of them, through all of them. Therefore, there are 2^3 = 8 total selections that can be made.

Now, I was going to model this problem with a multinomial distribution with 8 outputs, with a logit-normal covariance on the prior probabilities, but after thinking about the problem for a while, there is a LOT of correlation that we know a priori about the selections. I would think we could hardcode this knowledge into the prior covariance and not use general LKJ priors.

My question: is this the right approach? Can anyone comment on prior work in this area (all of my searches have come up empty). Any/all ideas or comments are super welcome!