I have some binary data. I have a proposed partition of this data into partitions 1 and 2.
I want to test whether the data in models 1 and 2 were generated by two Bernoullis such that their parameters p1 and p2 differ by at least some threshold T, or both were generated by the same Bernoulli.
I am looking for a prior that would approximate this hard threshold but still be easy to work with computationally
Thanks
Choosing priors can be subjective when you don't know all the details of the problem, but this is what I would do:
Normally ignorance priors for Bernoulli parameters are chosen to be Beta distributions, with common choices being $\mathrm{Beta}(0,0)$, $\mathrm{Beta}(1/2,1/2)$, and $\mathrm{Beta}(1,1)$ (see wikipedia). Here you are going to have to condition on your knowledge that $\left|p_1-p_2\right|>T$. So I would let $p_1$ and $p_2$ be independent $\mathrm{Beta}(\alpha,\alpha)$ variables and then set the probability of the region where they are too close together equal to $0$, renormalizing elsewhere. Unless I had a complelling reason to do otherwise I'd pick $\alpha=1$ because otherwise this "cutoff" is going to lead to nasty integrals.
Then in the case where they have the same distribution I'd let the parameter $p$ have a $\mathrm{Beta}(\alpha,\alpha)$ with the same $\alpha$ that I'd chosen before.