2 The model

We consider $i=1,2,\dots,n$ subpopulation. The available data from which the inferences are to be drawn are vectors of gene (allele) frequencies in these subpopulations, at a number of different loci. Differences between local gene frequencies is assumed to be a result of local genetic drift balanced by migration (or mutation). The pattern of migration between these subpopulation is described by a migration matrix $\mathbf{M}$ with elements $m_{ij}$ denoting the probability, or the proportion, of genes which originates from subpopulation $j$, given migration to subpopulation $i$. The objective of analysis is to obtain an estimate of the migration matrix $\mathbf{M}$. In general, we will be interested in migration matrices of certain biologically interesting forms only, involving a small number of parameters, which we hope to be able to estimate. The effective size, or at least, the relative size, of each subpopulation, $N_i$, is treated as known parameters. All subpopulations are also assumed to receive a small amount of immigrant genes from a large `outside world' population with gene frequency $q$.

With these assumptions, one can show that the gene frequencies, at equilibrium, provided that the fluctuations around $q$ are small, have covariance matrix $\mathbf{C}$, satisfying the matrix equation

$$\mathbf{C} = \mathbf{M} \mathbf{C} \mathbf{M}^T + \mathbf{E},$$ (1)

where

$$e_{ij}=\left\{\begin{matrix}\frac{1}{2N_i} &\text{for } i=j \\ 0 &\text{for } i\neq j, \\ \end{matrix}\right.$$ (2)

Using the solution of (1), the idea is to compute the likelihood of the data approximately by assuming multivariate normality. It is then possible to obtain maximum likelihood estimates of the parameters of any model for the migration matrix using numerical methods.