Problems with determination of noncommunicating classes for Monte Carlo Markov chain applications in pedigree analysis

Exact calculations for probabilities on complex pedigrees are computationally intensive and very often infeasible. Markov chain Monte Carlo methods are frequently used to approximate probabilities and likelihoods of interest. However, when a locus with more than two alleles is considered, the underlying Markov chain is not guaranteed to be irreducible and the results of such analyses are unreliable. A method for finding the noncommunicating classes of the Markov chain would be very useful in designing algorithms that can jump between these classes. In this paper, we will examine some existing work on this problem and point out its limitations. We will also comment on the difficulty of developing a useful algorithm.