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456 Statistical geneticslocidata • j•••Y • ,j••imeiosesFIGURE 39.2Imputation and specification of inheritance of DNA. (a) Left: The MCMCstructure of meiosis and linkage (Thompson, 2000). (b) Right: The IBD graphspecifying DNA shared by descent among observed individuals (Thompson,2003).tered problems of non-irreducibility of samplers (Sheehan and Thomas, 1993)and other mixing problems (Lin et al., 1993). Significant improvements wereobtained by instead using the inheritance patterns of Figure 39.1(b) as latentvariables (Thompson, 1994). The aprioriindependence of meioses, theMarkov dependence of inheritance at successive loci, and the dependence ofobservable data on the inheritance pattern at a locus (Figure 39.2(a)) leadto a variety of block-Gibbs samplers of increasing computational efficiency(Heath, 1997; Thompson and Heath, 1999; Tong and Thompson, 2008).An additional aspect of these later developments is the change from use ofthe conditional independence structure of a pedigree to that of an IBD-graph(Figure 39.2(b)). This graph specifies the individuals who share genome identicalby descent (IBD) at a locus; that is, DNA that has descended to currentindividuals from a single copy of the DNA in a recent common ancestor. Theobserved trait phenotypes of individuals are represented by the edges of theIBD graph. The nodes of the graph represent the DNA carried by these individuals;the DNA types of the nodes are independent. Each individual’s edgejoins the two DNA nodes which he/she caries at the locus, and his/her traitphenotype is determined probabilistically by the latent allelic types of thesetwo nodes. In the example shown in Figure 39.2(b), individuals D, G, and Fall share IBD DNA at this locus, as represented by the node labeled 4. Also,individuals B and J share both their DNA nodes, while C carries two copies ofa single node. Computation of probabilities of observed data on a graph suchas that of Figure 39.2(b) is described by Thompson (2003). This approachgives a much closer parallel to graphical models in other areas of statisticalscience; see, e.g., Lauritzen (1996) and Pearl (2000).