Page 280 - Applied Probability

12. Models of Recombination
268
i+n+k−j −λt
/(i + n + k − j)! is the probability of passing through
e
u ns k (λt)
the i − 1 current o points to the right, hitting the next χ point, returning
to a χ point after encountering n further points, and then passing through
k − j remaining o points en route to a χ point k points down the road.
We are now in a position to calculate gamete probabilities. According to
formula (12.6), we must first calculate the avoidance probability

Pr N I i =0
i∈T
for a subset {I i : i ∈ T} of the ordered, adjacent intervals I 1 ,...,I k . Sup-
pose, for example, that k = 3 and T = {1, 3}. Let interval I i have length
x i . At the start of interval I 1 , the Poisson-skip process is in state r of the
first Markov chain with equilibrium probability π r . On the interval I 1 , the
process must not encounter a χ point. It successfully negotiates the interval
and winds up at state s with probability q rs (x 1 ). On interval I 2 , there is
no restriction on the process, so it moves from state s at the start of the
interval to state t at the end of the interval with probability p st (x 2 ). On
interval I 3 , the process again must not encounter a χ point. Therefore, the
process successfully ends in state u with probability q tu (x 3 ). Summing over
all possible states at the start and finish of each interval gives the avoidance
probability

=0) = π r q rs (x 1 )p st (x 2 )q tu (x 3 ).
Pr(N I 1 + N I 3
r s t u
In obvious matrix notation, this reduces to
=0) = πe x 1 Φ x 2 Γ x 3 Φ 1.
e
e
Pr(N I 1 + N I 3
The general case is handled in exactly the same fashion.
Avoidance probabilities can be combined to give a compact formula for
gamete probabilities by defining the two matrices
1 xΓ xΦ
R(x)= (e − e )
2
1 xΓ xΦ
Z(x)= (e + e ).
2
Returning to our special case with three intervals, suppose that we wish
to calculate y {1,2} . Applying the distributive law in the gamete probability
formula (12.6), we easily deduce that

= πR(x 1 )R(x 2 )Z(x 3 )1.
y {1,2}
In effect, we choose on each interval whether to avoid χ points, and thus
use matrix e xΦ , or whether to embrace both χ and o points, and thus use