Page 251 - Applied Probability

11. Radiation Hybrid Mapping
238
parameters takes either of the equivalent generic forms
E(#successes | obs,γ old )
=
γ new,i
E(#trials | obs,γ old )
γ old,i (1 − γ old,i ) ∂L(γ old )
= γ old,i + ∂γ i , (11.8)
E(#trials | obs,γ old )
where obs denotes the observations X over all clones, and L is the loglikeli-
hood function. The second form of the update (11.8) requires less thought
to implement since only mechanical differentiations are involved in form-
ing the score. If the number of clones is H, then H is also the expected
number of trials appearing in the denominator for both updates to θ i . The
expected number of trials for r coincides with the expected number of frag-
ments. This expectation can be found by letting N i be the random number
of breaks between loci i and i + 1 over all clones. The first form of the
update for θ i shows that
E(N i | obs,γ old )
θ new,i = .
H
It follows that the expected number of fragments over all H clones is
m−1 m−1

H + E(N i | obs,γ old )= H(1 + θ new,i ).
i=1 i=1

11.5 Application to Haploid Data

TABLE 11.1. Best Locus Orders for Haploid Radiation Hybrid Data
Orders ∆L Breaks
1 2 3 4 5 6 7 8 9 10 11 12 13 0.00 123
1 2 3 4 5 6 7 8 10 9 11 12 13 1.49 125
1 2 3 4 5 13 12 11 10 9 8 7 6 1.79 126
5 4 3 2 1 6 7 8 9 10 11 12 13 1.84 128
1 2 3 4 5 7 6 8 9 10 11 12 13 1.93 127
6 7 1 2 3 4 5 8 9 10 11 12 13 2.26 127
6 7 5 4 3 2 1 8 9 10 11 12 13 2.43 128
1 2 3 4 5 6 7 8 11 10 9 12 13 3.22 127
1 2 3 4 5 6 7 8 11 9 10 12 13 3.23 127
1 2 3 4 5 13 12 11 9 10 8 7 6 3.28 128