An Algorithm for the Conditional Distribution of Independent Binomial Random Variables Given the Sum

We investigate Metropolis–Hastings (MH) algorithms to approximate the distribution of independent binomial random variables conditioned on the sum. Let Xi∼BIN(ni,pi). We want the distribution of [X1,…,Xk] conditioned on X1+⋯+Xk=n. We propose both a random walk MH algorithm and an independence sampling MH algorithm for simulating from this conditional distribution. The acceptance probability in the MH algorithm always involves the probability mass function of the proposal distribution. For the random walk MH algorithm, we take this distribution to be uniform across all possible proposals. There is an inherent asymmetry; the number of moves from one state to another is not in general equal to the number of moves from the other state to the one. This requires a careful counting of the number of possible moves out of each possible state. The independence sampler proposes a move based on the Poisson approximation to the binomial. While in general, random walk MH algorithms tend to outperform independence samplers, we find that in this case the independence sampler is more efficient.