The recursive derivation of likelihoods on complex pedigrees

It is frequently required in human genetics to calculate the likelihood of a model given the phenotypes of the individuals within some pedigree. Lange and Elston (1975) have recently presented a recursive method which enables this calculation to be made for, in their terminology, simple pedigrees; those in which at least one of each married couple is an original. The basic methodology of Lange and Elston is to successively reduce the size of the pedigree by collapsing the phenotypic information available on individuals onto parents, and hence to proceed recursively through the pedigree. We shall call this process 'peeling'. By generalizing the recursion so that information on individuals can be collapsed onto an offspring we allow pedigrees in which there are no loops (zero-loop), by introducing the notion of collapsing information onto a set of individuals jointly, we extend the method to pedigrees of arbitrary complexity. We note in passing that zero-loop pedigrees are a smaller class than non-inbred pedigrees. This and other aspects of the method will be treated more exhaustively elsewhere.