We assume that there is a hidden allele frequency P whose exact distribution will not be important to us, but is diffuse across the unit interval (0,1). Then conditional on P we assume that p has mean P(1,1,…,1) and covariance matrix P(1 – P)B where B is independent of P. This is a natural framework, used (filling in details variously) by Balding and Nichols [44], Nicholson et al. [18], and STRUCTURE [9] in the correlated allele mode. For small population divergence, we can take the diagonal entry Bii as the divergence (FST) between P and pi. Set and assume that all τi are of order τ, which is small. Conditional on p, then the Cj are independent. Cj has mean p and variance 2p(1 − p) where p = pi (j). This assumes Hardy–Weinberg equilibrium in each of the K populations.
