Markov chain Monte Carlo (MCMC) methods have been explored by various researchers as an alternative to exact probability computation in statistical genetics. The objective is to simulate a Markov chain with the desired equilibrium distribution. If the transition kernel is aperiodic and irreducible, then convergence to the equilibrium distribution is guaranteed; realizations of the Markov chain can thus be used to estimate desired probabilities. Aperiodicity is easily satisfied, but, although it has been shown that irreducibility is satisfied for a diallelic locus, reducibility is a potential problem for a multiallelic locus. This is a particularly serious problem in linkage analysis, because multiallelic markers are much more informative than diallelic markers and thus highly preferred. In this paper, the authors propose a new algorithm to achieve irreducibility of the Markov chain of interest by introducing an irreducible auxiliary chain. The irreducibility of the auxiliary chain is obtained by assigning positive probabilities to a small subset of the genotypic configurations inconsistent with the data, to bridge the gap between the irreducible sets.
Algorithms Alleles