The identity by descent process along the chromosome

The probabilities of the various possible identity by descent (IBD) states at a locus captures all the genealogical information for that locus for the set of individuals under consideration. Here we study the stochastic process of the IBD state as one moves across the genome of a set of individuals. In general it is no longer sufficient to specify the IBD state, one needs to increase the state space if one is to maintain the Markov property, as has been discussed by for instance McPeak and Sun [Am J Hum Genet 2000;66:1076-1094] and Browning and Browning [Theor Popul Biol 2002;62:1-8]. This paper discusses a general method of deriving the transition matrix for that Markov chain iteratively from one time point to a subsequent one. This method allows a considerable reduction in the size of the state space needed. The basic recursion is set out here and the application is illustrated by two specific examples.