Page 281 - Applied Probability

12. Models of Recombination
269
xΓ
matrix e . The factors of
overall factor ( ) in (12.6), and the factors of +1 and −1 make the sign
2
come out right. In general, if a set S is characterized by the
(−1)
|S∩T |
k-tuple of indicators s i =1 {i∈S} , then the gamete probability y S can be
expressed as 1 k 1 2 in the definition of R(x) and Z(x) give the
= πR(x 1 ) Z(x 1 ) 1−s 1 ··· R(x k ) Z(x k ) 1−s k 1. (12.14)
s 1
s k
y S
Formula (12.14) can also be derived and implemented by defining an ap-
propriate hidden Markov chain. Let U j be the unobserved state of the first
Markov chain at locus j, and let Y j be the observed indicator random vari-
able flagging whether recombination has occurred between loci j −1 and j.
To compute a gamete probability Pr(Y 2 = i 1 ,...,Y k+1 = i k ), one can ap-
ply Baum’s forward algorithm as in the radiation hybrid model of Chapter
11 [2, 7]. We begin the recursive computation of the joint probabilities
f j (u j ) = Pr(Y 2 = i 1 ,... ,Y j = i j−1 ,U j = u j )
. At locus k + 1 we recover the gamete probability
by setting f 1 (u 1 )= π u 1
Pr(Y 2 = i 1 ,...,Y k+1 = i k ) from the identity

Pr(Y 2 = i 1 ,...,Y k+1 = i k )= f k+1 (u k+1 ).
u k+1
In view of Mather’s formula (12.1), if i j = 1, then
1
f j+1 (u j+1 )= f j (u j ) p u j ,u j+1 (x j ) − q u j ,u j+1 (x j )
2
u j
1
2
because [p u j ,u j+1 (x j )−q u j ,u j+1 (x j )] is the probability that the chain moves
from state U j = u j at locus j to state U j+1 = u j+1 at locus j + 1 and that
the chosen gamete is recombinant on the interval between the loci. On the
other hand, if i j = 0, then
1
f j+1 (u j+1 )= f j (u j ) p u j ,u j+1 (x j )+ q u j ,u j+1 (x j )
2
u j
because
1
(x j )+ (x j )
q u j ,u j+1 p u j ,u j+1 (x j ) − q u j ,u j+1
2
1
= p u j ,u j+1 (x j )+ q u j ,u j+1 (x j )
2
is the probability that the chain moves from state U j = u j at locus j to state
U j+1 = u j+1 at locus j + 1 and that the chosen gamete is nonrecombinant
on the interval between the loci. Hence, Baum’s forward algorithm is simply
a device for carrying out the vector times matrix multiplications implied
by formula (12.14).