Efficient computation for Bayesian comparison of two proportions

In Bayesian comparison of two proportions, the exact computation of the evidence involves evaluating a generalized hypergeometric function. Several agreeing, but not identical, expressions for the evidence have been derived in the literature; however, their practical computation (by summing the truncated hypergeometric series) can be troubled by slow convergence or catastrophic cancellation. Using a set of equivalence relations for the generalized hypergeometric function, we derive ten equivalent expressions for the evidence: We show that one of these formulations, which has not previously been studied, is superior in terms of its computational properties. We recommend that this be used instead of existing formulations, and provide an efficient software implementation.