Gene extinction and allelic origins in complex genealogies

With the increasing emphasis on data analysis in mathematical genetics, problems of parametrizing genealogical structure become of practical importance. A complete specification of the genetic effects of genealogical structure is provided by the probabilities of genetically distinct states of gene identity by descent. Although this provides a direct parametrization for the joint distribution of traits on a set of related individuals, it is an unwieldy tool in the analysis of large and complex genealogies. Probabilities of joint descent of founder genes and likely ancestries of alleles provide alternative characterizations of relationship and have direct application in practical problems. Joint extinction probabilities of founder genes can also be derived as ancestral likelihoods: evolutionarily, the most significant characteristic of a genealogical structure must be its effect on the survival and extinction of genes.