With nested models, as in the HWE context, an appealing specification is one under which π(θ | H0) = π(θ | H1), so that when comparing the data under the null and alternative models we are examining whether the addition of the extra set of parameters that define the alternative leads to a better explanation of the data. This relationship does not hold under conjugate specifications for the null and alternative, but does hold for the following specification for k = 2. Under the alternative suppose we assume π1(p11, p12, p22) is Dir(1,1,1). To obtain a consistent prior under the null we reparameterize from (p11, p12, 1 – p11 – p12) to (p1, p12, 1 – p1 p12/2) and then integrate over p12 to give: (8)π0(p1)={4p10<p1≤0.54(1−p1)0.5<p1<1.} For these priors the Bayes factor is again available in easily computable form with the denominator as in the conjugate case, i.e., equation (7) with k = 2 and v = (1, 1, 1), and p(n∣H0)=n!2n12+2n11!n12!n22![IBe(0.5,2n11+n12+2,n12+2n22+1)+Be(2n11+n12+1,n12+2n22+2)−IBe(0.5,2n11+n12+1,n12+2n22+2)], where Be(a, b) = Γ(a)Γ(b)/Γ(a + b) and IBe(x,a,b)=∫0xza−1(1−z)b−1dz is the incomplete beta function (which is straightforward to
