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

Markov chain Monte Carlo (MCMC) methods have been explored byvarious researchers as an alternative to exact probability computationin statistical genetics. The objective is to simulate a Markovchain with the desired equilibrium distribution. If the transitionkernel is aperiodic and irreducible, then convergence to theequilibrium distribution is guaranteed; realizations of theMarkov chain can thus be used to estimate desired probabilities.Aperiodicity is easily satisfied, but, although it has beenshown 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 thandiallelic markers and thus highly preferred. In this paper,the authors propose a new algorithm to achieve irreducibilityof the Markov chain of interest by introducing an irreducibleauxiliary chain. The irreducibility of the auxiliary chain isobtained by assigning positive probabilities to a small subsetof the genotypic configurations inconsistent with the data,to bridge the gap between the irreducible sets.