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11. Radiation Hybrid Mapping
248
prior. This suggests that the EM gradient update and the ordinary EM
update for r will be equally effective in finding the posterior mode.
From the Bayesian perspective, perhaps more important than finding
the posterior mode is the possibility of computing posterior probabilities
for the various locus orders. Under the natural assumption that all orders
are a priori equally likely, the posterior probability of a given order α is
" L α (γ)+R α(γ)
e dγ
, (11.20)
" L β (γ)+R β (γ)
e dγ
β
where the sum in the denominator ranges over all possible orders β and
L β and R β denote the loglikelihood and log prior appropriate to order β.
Two ugly issues immediately rear their heads at this point. First, unless the
number of loci m is small, the number of possible orders can be astronom-
ical. This problem can be finessed if the leading orders can be identified
and the sum truncated to include only these orders. In many problems only
a few orders contribute substantially to the denominator of the posterior
probability (11.20).
The other issue is how to evaluate the integrals appearing in (11.20). Due
to the complexity of the integrands, there is no obvious analytic method
of carrying out the integrations. For haploid data, Lange and Boehnke
[14] suggest two approximate methods. Both of these methods have their
drawbacks and can be computationally demanding. Here we suggest an
approximation based on Laplace’s method from asymptotic analysis [7, 22].
The idea is to expand the logarithm of the integrand e L α (γ)+R α(γ) in a
second-order Taylor’s series around the posterior mode ˆ γ. Recalling the
well-known normalizing constant for the multivariate normal density and
defining F α (γ)= L α (γ)+ R α (γ), this approximation yields

1
t 2
γ
γ
γ
γ
e F α(γ) dγ ≈ e F α (ˆ)+ (γ−ˆ) d F α (ˆ)(γ−ˆ) dγ
2
m
γ
2
= e F α (ˆ) (2π) 2 det(−d F α (ˆ γ)) − 1 2 . (11.21)
The accuracy of Laplace’s approximation increases as the log posterior
function becomes more peaked around the posterior mode ˆ γ. The quadratic
2
form d F α (ˆ γ) measures the curvature of F α (γ)at ˆ γ.
11.10 Application to Diploid Data
Table 11.2 lists the 10 best orders identified for 6 loci on chromosome 4
from 85 diploid clones created at the Stanford Human Genome Center and
distributed by Research Genetics of Huntsville, Alabama. These six se-
quence tagged sites constitute a small subset of a much more extensive
set of chromosome 4 markers. The columns labeled Prob. 1, Prob. 2, and