(A1)Hey et al. 10.1073/pnas.0611164104. |
SI Text
The Isolation with Migration Model. The parameters of the two-population isolation-with-migration model include a population mutation rate for each of the three populations (q1, q2 and qa, where qa is the population size parameter for the ancestral population), the scaled time at which the ancestral population gave rise to the two descendant populations, t, and the two scaled migration rates, m1 and m2 . The application to this model follows closely on that for the two-population island model. Let n1,t and n2,t be the number of gene copies in G in populations 1 and 2 at time t, respectively . Before t, there will be a total of aB = n1 - n1,t + n2 - n2,t + w1 + w2 + 1 time intervals, and after t there will be aA= n1,t + n2,t - 1 time intervals.
Letting Θ={
q1, q2 ,qa, m1,m2}, the probability of a genealogy for a given set of parameter values is (A1)
Where the f and g terms refer to the total coalescent and migration rates, respectively over the corresponding portions of G, as for the two population island model, such that
(A2)
The prior probability of G and t is
(A3)
Integration yields a product of five terms, including three that take the same form as (12) and two that take the same form as (18).
Finally
(A4)
where the numerator is the product of (A1) and the prior distribution and the denominator is given by (A3). With a sample of k genealogies, (A4) can be approximated as
. (A5)
Multiple Loci.
For a data set from j independently segregating loci, let X= {X1, X2, . . . Xj} and G={G1, G2, . . . Gj}. The model parameters are unchanged from the single locus case with the exception that the mutation rate by which each is scaled is the geometric mean of the mutation rates of all of the loci . The variation among loci is accommodated by including as parameters a set of j mutation rate scalars, u = {u1, u2, . . . uj}, such that
. Unlike the components of Θ, we cannot integrate over the mutation rate scalars and must include them in the Markov chain and record sets of values {G,u} over the course of the simulation. However it remains possible to generate samples from the posterior density for G, because it is possible to calculate 
(A6)