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Simultaneous sampling of the pedigree was achieved by sampling descent graphs using the Metropolis-Hastings algorithm.
Sampling descent graphs overcomes most, if not all, of the limitations incurred by iterative peeling algorithms.
Figure 1 shows a pedigree and one of its many possible descent graphs.
The generating function actually explicitly lists the numbers of paths of any lengths over all possible descent graphs of a pedigree.
Given a i and b i are IBD, there may be many possible inheritance paths from different descent graphs connecting them.
Notice that there are only finite number of descent graphs for a given pedigree; therefore, there are only finite number of inheritance paths between two alleles and the summation only has finite number of terms.
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To derive our model, we first introduce the concept of descent graph and define an inheritance-generating function between a pair of alleles in a descent graph.
Each descent graph specifies how each of the founding alleles is inherited.
A descent graph illustrates one possible inheritance pattern within a pedigree, and by definition, it does not include genotype information.
A descent graph describes the inheritance state of each allele and provides pedigrees guaranteed to be consistent with Mendelian sampling.
The vector of selectors, which is called the inheritance vector, can get 22 n assignments and each assignment s defines a descent graph denoted by D[ s].
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