In the diallelic case the conjugate priors Dir(w1, w2) and Dir(v11, v12, v22) under the null and alternative lead to the Bayes factor: (5)BF=2n12Γ(w)Γ(2n11+n12+w1)Γ(v11)Γ(v12)Γ(v22)Γ(n12+2n22+w2)Γ(n+v)Γ(w1)Γ(w2)Γ(2n+w)Γ(v)Γ(n11+v11)Γ(n12+v12)Γ(n22+v22), where w = w1 + w2 and v = v11 + v12 + v22 (Consonni et al., 2008). For k alleles and under conjugate priors the normalizing constants for the HWE and saturated models are available in closed form and are given by: (6)p(n∣H0)=n!2∑i,j=1,j>iknij∏i,j=1,j≥iknij!Γ(w)∏i=1kΓ(wi)×∏i=1kΓ(wi+2nii+∑j>inij)Γ(w+∑i=1k2nii+∑i=1k∑j>inij),(7)p(n∣H1)=n!∏i,j=1,j≥inij!Γ(v)∏i,j=1,j≥iΓ(vij)×∏i,j=1,j≥iΓ(vij+nij)Γ(v+n), where w=∑i=1kwi and v=∑i,j=1,j≥ikvij. The ratio of equation (6) to equation (7) gives the Bayes’ factor, of which (2) is the special case when k = 2.
