Approximate Bayes Factors and Orthogonal Parameters, with Application to Testing Equality of Two Binomial Proportions

SUMMARY We use asymptotic expansions to approximate Bayes factors, improving on a method used by Jeffreys. Suppose that the hypothesis H 0: ψ = ψ0 is to be tested against H A: ψ ≠ ψ0 in the presence of a nuisance parameter β, and initially priors π0(β) under H 0 and π(β, ψ) under H A are used. We consider the problem of assessing sensitivity of the Bayes factor to small changes in π0 and π. We show that for local alternatives (which, for moderate sample sizes, are consistent with small or moderately large values of the Bayes factor in favour of the alternative), if β and ψ are what we call ‘null orthogonal’ parameters, then alterations in π0 have no effect on the Bayes factor up to order O(n –1). Under similar conditions we also derive an order O(n –1) approximation to the minimum Bayes factor over all priors π under H A such that the marginal prior on ψ is normal with mean ψ0. We then go on to consider sensitivity to specific changes in the marginal prior on ψ and show how asymptotics may be used for this, applying a second-order approximation due to Tierney and Kadane. We illustrate the results with a test of equality of two binomial proportions and briefly investigate the accuracy of the approximations is this context.