Page 199 - Applied Probability

9. Descent Graph Methods
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graph C without passing through an illegal descent graph such as descent
graph B, where the great-grandchild inherits a homozygous genotype.
The remedy to this dilemma is to “tunnel through” illegal descent graphs
by taking multiple transitions per step of the Markov chain [32]. In practice,
we employ a random number of transitions per step of the chain. This per-
mits the chain to pass through illegal descent graphs on its way between
legal descent graphs. Among the many devices for selecting the random
number of transitions per step, one of the most natural is to sample from
a geometric distribution with mean 2. This procedure entails taking a sin-
1
1
gle transition with probability , two transitions with probability , three
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2
transitions with probability 1 , and so forth. For each transition one ran-
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domly selects a transition rule and a person and locus. If the transition rule
selected is T 0, then one also randomly selects a maternal or paternal node
to switch. If one of the T 2 transitions is selected, then one also randomly
selects a spouse of the selected person.
Although selections are random, they need not be uniform. For example,
it is probably wise to select transition T 0 more often than T 1 , and T 1 more
often than T 2 , and to target untyped people more often than typed people.
It makes sense to make other choices uniformly, such as the selection of
a spouse for a T 2 transition or of a maternal or paternal node for a T 0
transition. In implementing the Metropolis algorithm, it simplifies matters
to keep the proposal distribution symmetric and use equation (9.6). This is
possible if independent choices are made at each transition. Indeed, because
each transition is its own inverse, taking a given sequence of transitions in
reverse order leads back from a proposed descent graph to the current
descent graph.
The Metropolis algorithm always takes steps to more favorable descent
graphs but never allows steps to illegal descent graphs. It is perfectly pos-
sible for the Markov chain to remain in place if a step is rejected or the
step consists of a double application of the same transition. This feature
forces the chain to be aperiodic. Finally, the chain is also irreducible since
the tunneling mechanism permits the chain to move from any legal descent
graph to any other legal descent graph in a single step.

9.7 Computing Location Scores

Location scores can be computed by a hybrid of stochastic sampling of
marker descent graphs and deterministic likelihood evaluation. Denote the
unknown trait position by d and the observed trait phenotypes on a pedi-
gree by T. Since it is trivial to compute the likelihood Pr(T) of the trait
phenotypes in the absence of the marker phenotypes, the key ingredient
in computing a location score log [Pr d (T | M)/ Pr(T)] is the conditional
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probability Pr d (T | M).