Page 247 - Applied Probability

11. Radiation Hybrid Mapping
234
2
1
5
4
3
6
I σ(1)

I σ(2)

I σ(3)

I σ(4)

I σ(5)

FIGURE 11.1. An Interval Match for the Permutation σ =(4, 6, 2, 1, 3, 5).

θ i,i+1 is the breakage probability between the two loci. Thus,
m−1

E[B 1 (id)] = 2r(1 − r) θ i,i+1 . (11.2)
i=1
The corresponding expression for an arbitrary permutation σ is
m−1

E[B 1 (σ)] = 2r(1 − r) θ σ(i),σ(i+1) . (11.3)
i=1
The interval I σ(i) defined by a pair {σ(i),σ(i +1)} is a union of adjacent
intervals from the correct order 1,... ,m. It is plausible to conjecture that
we can match in a one-to-one fashion each interval (k, k + 1) against a
union I σ(i) containing it. See Figure 11.1 for a match involving the permu-
tation (σ(1),σ(2),σ(3),σ(4),σ(5),σ(6)) = (4, 6, 2, 1, 3, 5). If this conjecture
is true, then either θ k,k+1 = θ σ(i),σ(i+1) when the union I σ(i) contains a sin-
gle interval, or θ k,k+1 <θ σ(i),σ(i+1) when the union I σ(i) contains several
intervals. If the former case holds for all intervals (k, k + 1), then σ = id.
The inequality E[B 1 (id)] < E[B 1 (σ)] for σ = id now follows by taking the
indicated sums (11.2) and (11.3).