Page 192 - Applied Probability

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9. Descent Graph Methods
of a prior probability and a transmission probability.
Under the usual assumptions of genetic equilibrium, Prior(G) is the prod-
uct of the population frequencies of the founder alleles involved in G. Since
a descent state entails no ambiguities about recombination, Trans(G) re-
duces under Haldane’s model of recombination to a product of a power of
2 1
and relevant powers of the recombination fractions and their complements
for the adjacent intervals separating the markers. Finally, owing to the fact
that all compatible descent states G ø G exhibit the same transmission
0
pattern, we can reexpress the likelihood (9.8) as

Pr(G ∩ M) = Trans(G) Prior(G). (9.9)
0
0
G à 0 G∩M
In the next section we tackle the subtle problem of quick computation of
the sum of priors Prior(G) [19, 34].
G à 0 G∩M
9.5 Descent Trees and the Founder Tree Graph
Given l loci in a pedigree with p people and f founders, there are 2lp nodes
in a descent graph. These nodes are grouped in 2lf descent trees. The
descent tree rooted at a particular founder node contains that founder node
and those non-founder nodes inheriting the corresponding founder gene. All
nodes of a descent tree involve the same locus. When a founder gene is not
passed to any descendant of the founder, then the descent tree exists but
is degenerate.
It is convenient to proceed to a higher level of abstraction and make
the founder trees into an undirected graph. This abstraction serves to keep
track of how founder alleles are constrained in a coupled manner by the ob-
served marker phenotypes in the pedigree. The nodes of the founder tree
graph are the descent trees of the descent graph. Two nodes of the founder
tree graph are connected by an edge if and only if the two corresponding
descent trees pass through the same typed locus of some person in the
pedigree. This definition precludes connecting two descent trees associated
with different loci. Part (a) of Figure 9.2 shows a descent graph for a single
marker locus in which each descent tree is labeled above its rooting founder
gene. Do not confuse these labels with the allele symbols used in descent
states. Part (b) of Figure 9.2 shows the founder tree graph corresponding
to this descent graph, assuming all nonfounders and no founders are typed
at the locus.
It is possible for two descent trees at the same locus to mutually impinge
on more than one person typed at the locus. Although this information
is relevant to discerning whether the two trees are genetically compatible
with the observed phenotypes in the pedigree, for the sake of simplicity, we
will still view the descent trees as connected by just a single edge. When