Page 258 - Applied Probability

11. Radiation Hybrid Mapping
245
Taking into account the defining condition for an obligate break leads to
the recurrence relation

p k (l, j)t c,k (l, i)+
p k (l, j − 1)t c,k (l, i)
p k+1 (i, j)=
l∼i
l ∼i
for all 0 ≤ i ≤ c, where l ∼ i indicates that l and i are simultaneously in ei-
ther the set {0} or the set {1,... ,c}, and where the transition probabilities
t c,k (l, i) are defined in (11.13). As already noted in the haploid case, when
the final locus k = m is reached, the probabilities p m(i, j) can be summed
on i to produce the distribution of the number of obligate breaks.
11.9 Bayesian Methods
Bayesian methods offer an attractive alternative to maximum likelihood
methods. To implement a Bayesian analysis of locus ordering, two technical
hurdles must be overcome. First, an appropriate prior must be chosen. Once
this choice is made, efficient numerical schemes for estimating parameters
and posterior probabilities must be constructed.
It is more convenient to put a prior on the distances between the adja-
cent loci of an order than on the breakage probabilities determined by these
distances. In designing a prior for interlocus distances, we can assume with
impunity that the intensity of the breakage process satisfies λ =1.It is
also reasonable to assume that the m loci to be mapped are sampled uni-
formly from a chromosome interval of known physical length. This length
may be difficult to estimate in base pairs. Furthermore, physical distances
measured in base pairs are less relevant than physical distances measured
in expected number of breaks (Rays). We can circumvent the calibration
problem of converting from one measure of physical distance to the other
by using the results of a maximum likelihood analysis. Suppose that un-
der the best maximum likelihood order, we estimate a total of b expected
breaks between the first and last loci. With m uniformly distributed loci,
adjacent pairs of loci should be separated by an average distance of b .
m−1
This quantity should also approximate the average distance from the left
end of the interval to the first locus and from the right end of the interval
(m+1)b
to the last locus. These considerations suggest that d = would be
m−1
a reasonable expected number of breaks to assign to the prior interval. In
practice, this value of d may be too confining, and it is probably prudent
to inflate it somewhat.
Given a prior interval of length d,let δ i be the distance separating the
adjacent loci i and i + 1 under a given order. To calculate the joint dis-
tribution of the vector of distances (δ 1 ,...,δ m−1 ), expand this vector to
include the distance δ 0 separating the left end of the interval from the first
locus. These spacings are related to the positions t 1 ,...,t m of the loci on