Page 260 - Applied Probability

to its left argument. In effect, we update γ by one step of Newton’s method
applied to the function Q(γ | γ old )+ R(γ), keeping in mind the identity
10
d Q(γ old | γ old )= dL(γ old ) proved in Problem 9 of Chapter 2.
All of the terms appearing in (11.19) are straightforward to evaluate. For
instance, taking into account relation (11.1) with λ =1, we have
∂
∂ 11. Radiation Hybrid Mapping 247
dθ i
L(γ)= L(γ)
∂δ i ∂θ i dδ i
∂
= L(γ)(1 − θ i ).
∂θ i
Differentiation of the log prior produces
∂ 1
R(γ)= −
∂δ i d − δ 1 −· · · − δ m−1
∂
R(γ)= 0
∂r
t
2
−d R =(dR) dR.
20
Computation of the diagonal matrix d Q(γ | γ) is more complicated.
Let N i be the random number of chromosomes in the sample with breaks
between loci i and i + 1. As noted earlier, this random variable has a
binomial distribution with success probability θ i and cH trials. Because
of the nature of the complete data likelihood, it follows that modulo an
irrelevant constant,
Q(γ | γ old)
=E(N i | obs,γ old)ln(θ i ) + E([cH − N i ] | obs,γ old ) ln(1 − θ i ).

Straightforward calculations show that
∂ 2 E(N i | obs,γ old )(1 − θ i )
∂δ 2 Q(γ | γ old )= − θ 2 .
i i
˜
If θ new,i is the EM update of θ i ignoring the prior, then as remarked pre-
˜
viously, E(N i | obs,γ old)= cHθ new,i .
It is possible to simplify the EM gradient update (11.19) by applying the
Sherman-Morrison formula discussed in Chapter 3. In the present context,
t −1
we need to compute (A + uu ) v for the diagonal matrix
20
A = −d Q(γ old | γ old )
t
and the vectors u = dR(γ old ) and v = dL(γ old )+ dR(γ old). Because R(γ)
∂
does not depend on r, the partial derivative ∂r R(γ) vanishes. Thus, the
t
matrix A + uu is block diagonal, and the EM gradient update for the
parameter r coincides with the EM gradient update for r ignoring the