On the Bayesian analysis of a non-identifiable Binomial model

We provide analysis for the posterior distribution and expectation of $(p_1,p_2)$ where $Y\mathrm{\mid }{p}_1,p_2\sim \text{Binomial}(n,p_1{p}_2)$ and $(p_1,p_2)$ is uniformly distributed on the unit square $[0,1{]}^2$. We exhibit interesting expressions in terms of a truncated Beta distribution, a finite mixture of Beta distributions and harmonic numbers, and derive a simple large sample size $n$ approximation for the posterior expectations $\mathbb{E}(p_i\mathrm{\mid }y)$ as well as for the normalization constant in the posterior joint density.