For the single f model, and with the prior p = (p1 , . . . , pk) ~ Dir(w) under the null, we may use importance sampling to evaluate the denominator of the Bayes factor (4). To obtain an efficient proposal we parameterize in terms of the set (θ, λ), where θ = (θ1 , . . . , θk–1) with θi = log(pi/pk), i = 1 , . . . , k – 1. Under this parameterization the restrictions on p and the awkward constraints on f|p are automatically satisfied, and (θ, λ) are defined on Rk, which is desirable for finding an efficient importance sampling proposal. Under the alternative we have p ~ Dir(w) multiplied by equation (9) and so require (10)p(n∣H1)=∫p(n∣p,f)πp(p)πf∣p(f∣p)dpdf=Eπpπf∣p[p(n∣p,λ)](11)=∫p(n∣θ,λ)πp(s(θ))πf∣p(t(θ,λ)∣p)∣J∣gθ,λ(θ,λ)×gθ,λ(θ,λ)dθdλ=Egθ,λ[p(n∣θ,λ)πp(h(θ))∣J∣πλ(λ∣p)gθ,λ(θ,λ)], where p = s(θ) and f = t(θ, λ) are the reverse transformations and J is the Jacobian for the transformation from (p, f) → (θ, λ). Details are contained in the Appendix. For small numbers of alleles, we sample directly from the prior as in (10), which is a computationally simple approach (because the
