An algorithm to sample unobservable genotypes in complex pedigrees

Probability functions such as likelihoods and genotype probabilities play an important role in the analysis of genetic data. When genotype data are incomplete Markov chain Monte Carlo (MCMC) methods, such as the Gibbs sampler, can be used to sample genotypes at the marker and trait loci. The Markov chain that corresponds to the scalar Gibbs sampler may not work due to slow mixing. Further, the Gibbs chain may not be irreducible when sampling genotypes at marker loci with more than two alleles. These problems do not arise if the genotypes are sampled jointly from the entire pedigree. When the pedigree does not have loops, a joint sample of the genotypes can be obtained efficiently via modification of the Elston-Stewart algorithm. When the pedigree has many loops, obtaining a joint sample can be time consuming. We propose a method for sampling genotypes from a pedigree so modified as to make joint sampling efficient. These samples, obtained from the modified pedigree, are used as candidate draws in the Metropolis-Hastings algorithm.