Coalescent theory is a model of how alleles sampled from a population may have originated from a common ancestor. In the simplest case, coalescent theory assumes no recombination, no natural selection, and no gene flow or population structure, meaning that each variant is equally likely to have been passed from one generation to the next. The model looks backward in time, merging alleles into a single ancestral copy according to a random process in coalescence events. Under this model, the expected time between successive coalescence events increases almost exponentially back in time (with wide variance). Variance in the model comes from both the random passing of alleles from one generation to the next, and the random occurrence of mutations in these alleles. The mathematical theory of the coalescent was developed independently by several groups in the early 1980s as a natural extension of classical population genetics theory and models, but can be primarily attributed to John Kingman. Advances in coalescent theory include recombination, selection, overlapping generations and virtually any arbitrarily complex evolutionary or demographic model in population genetic analysis. The model can be used to produce many theoretical genealogies, and then compare observed data to these simulations to test assumptions about the demographic history of a population. Coalescent theory can be used to make inferences about population genetic parameters, such as migration, population size and recombination. Theory Time to coalescence Consider a single gene locus sampled from two haploid individuals in a population. The ancestry of this sample is traced backwards in time to the point where these two lineages coalesce in their most recent common ancestor (MRCA). Coalescent theory seeks to estimate the expectation of this time period and its variance. The probability that two lineages coalesce in the immediately preceding generation is the probability that they share a parental DNA sequence. In a population with a constant effective population size with 2Ne copies of each locus, there are 2Ne "potential parents" in the previous generation. Under a random mating model, the probability that two alleles originate from the same parental copy is thus 1/(2Ne) and, correspondingly, the probability that they do not coalesce is 1 − 1/(2Ne). At each successive preceding generation, the probability of coalescence is geometrically distributed—that is, it is the probability of noncoalescence at the t − 1 preceding generations multiplied by the probability of coalescence at the generation of interest: For sufficiently large values