Achieving irreducibility of the Markov chain Monte Carlo method applied to pedigree data

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.