Sampling genotypes on complex pedigrees with phenotypic constraints: the origin of the B allele among the Polar Eskimos.

Exact probability calculations are often infeasible on large complex pedigrees. Conditional independences, however, occurring as a natural consequence of Mendelian inheritance of genetic traits, define a locally dependent Markov random field on the state space of all genotypic configurations on the pedigree. The underlying Markov chain is irreducible for most traits determined by a diallelic locus. For a given pedigree and a known genetic model, the Gibbs sampler can be used to obtain good estimates of the posterior distribution of genotypes given the observed data. The areas of pedigree analysis to which such an approach would be most directly relevant include genetic counselling and selective animal breeding, together with questions about ancestral genotypes and the ancestral paths of rare alleles. The method is illustrated by tracing the ancestral paths of a rare allele in a simple diallelic system on a highly complex Eskimo pedigree.