Page 262 - Applied Probability

11. Radiation Hybrid Mapping
249
Prob. 3 are posterior probabilities calculated under various approximations
L β (γ)+R β (γ)
"
dγ in formula (11.20). The first approxima-
of the integrals
e
tion is

γ
γ
L β (γ)+R β (γ)
L β (ˆ)+R β (ˆ)
e
∝ e
dγ
,
(11.22)
where ˆ γ is the maximum likelihood estimate and the log prior function
R β (γ) is taken as 0. The second approximation uses the actual log prior
function in (11.22) and replaces the maximum likelihood estimate by the
posterior mode. The third approximation is just the Laplace approximation
(11.21). For the numbers shown in Table 11.2, all 360 = 6! orders were
2
included in the denominator of (11.20).
The three posterior probabilities displayed in Table 11.2 evidently agree
well. Except for one minor reversal for the Laplace approximation, the 10
listed orders have the same ranks. These posterior probability ranks are
roughly similar to the ranks based on minimum obligate breaks.
TABLE 11.2. Best Locus Orders for Diploid Radiation Hybrid Data
Orders Prob. 1 Prob. 2 Prob. 3 Breaks
1 2 3 4 5 6 .36114 .35690 .34569 52
1 2 3 5 4 6 .32028 .33051 .32845 51
2 3 4 5 6 1 .16736 .16301 .16601 51
2 3 5 4 6 1 .14554 .14451 .15400 51
1 2 3 6 5 4 .00244 .00222 .00233 54
1 4 5 6 3 2 .00136 .00119 .00128 54
1 3 2 4 5 6 .00054 .00045 .00053 56
1 2 3 5 6 4 .00038 .00036 .00054 54
1 3 2 5 4 6 .00024 .00022 .00029 55
1 4 6 5 3 2 .00021 .00019 .00029 54
The failure of Table 11.2 to identify a decisively best order reflects uncer-
tainties in placing locus 1 to the right or to the left of the major cluster of
loci and in reversing loci 4 and 5 in this cluster. The maximum likelihood
odds for pair reversals under the best identified order are given in the dia-
gram below. It is interesting that the odds for inverting loci 1 and 2 provide
no hint of the overall ambiguity in ordering locus 1. Clearly, caution should
be exercised in interpreting pairwise inversion odds.

Pairwise Inversion Odds for the Best Order

5.8×10 9 6.7×10 2 1.7×10 13 1.1 1.5×10 4
1 —— 2 —— 3 —— 4 —— 5 —— 6
Maximum likelihood estimates of the interlocus distances under the best
order are as follows: