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E.A. Thompson 453PopulationMeiosisG (F )M3G (F )TG (F )M2G (F )M1FoundersS M3S TS M2S M1Joint TransmissionG M3G TG M2G M1AllHaplotypesY M3Y TTraitPenet.Y M2Y M1MarkerPenet.TraitPenet.MarkerErrorA (F )M3 A (F )T A (F )M2 A (F )M1Y M3Y TY M2Y M1MarkerPop nTraitPop nMarkerPop nMarkerPop nFIGURE 39.1The two orthogonal conditional independence structures of genetic data on relatedindividuals. (a) Left: The conditional independence of haplotypes amongindividuals. (b) Right: The conditional independence of inheritance amonggenetic loci. Figures from Thompson (2011), reproduced with permission ofS. Karger AG, Basel.of descendant individuals. Finally, penetrance parameters specify the probabilisticrelationship between these genotypes and observable genetic data (Y )on individuals, again at each trait (T ) or marker (M) locus.Thesepenetrancemodels can incorporate typing error in marker genotypes, as well asmore complex relationships between phenotype and genotype. The structuredparametric models of statistical genetics lead naturally to likelihood inference(Edwards, 1972), and it is no accident that from Fisher (1922b) onwards, amajor focus has been the computation of likelihoods and of maximum likelihoodestimators.Methods for the computation of likelihoods on pedigree structures makeuse of the conditional independence structure of genetic data. Under the lawsof Mendelian genetics, conditional on the genotypes of parents, the genotypesof offspring are independent of each other, and of those of any ancestral andlateral relatives of the parents. Data on individuals depends only on their genotypes;see Figure 39.1(a). Methods for computation of probabilities of observeddata on more general graphical structures are now widely known (Lauritzenand Spiegelhalter, 1988), but these methods were already standard in pedigreeanalysis in the 1970s. In fact the first use of this conditional independence incomputing probabilities of observed data on three-generation pedigrees datesto Haldane and Smith (1947), while the Elston–Stewart algorithm (Elston and