Recurrence Equations for the Probability Distribution of Sample Configurations in Exact Population Genetics Models

Recurrence equations for the number of types and the frequency of each type in a random sample drawn from a finite population undergoing discrete, nonoverlapping generations and reproducing according to the Cannings exchangeable model are deduced under the assumption of a mutation scheme with infinitely many types. The case of overlapping generations in discrete time is also considered. The equations are developed for the Wright–Fisher model and the Moran model, and extended to the case of the limit coalescent with nonrecurrent mutation as the population size goes to∞ and the mutation rate to 0. Computations of the total variation distance for the distribution of the number of types in the sample suggest that the exact Moran model provides a better approximation for the sampling formula under the exactWright–Fisher model than the Ewens sampling formula in the limit of the Kingman coalescent with nonrecurrent mutation. On the other hand, this model seems to provide a good approximation for a -coalescent with nonrecurrent mutation as long as the probability of multiple mergers and the mutation rate are small enough.