Page 254 - Applied Probability

Preliminary estimates of the parameters r and θ can be derived from the
empirically observed values of Pr(X 1 =0,X 2 = 0) and Pr(X 1 =1,X 2 = 1).
In fact, the equation
Pr(X 1 =1,X 2 =1) − Pr(X 1 =0,X 2 = 0)=1 − 2(1 − r)
can be solved to give 11. Radiation Hybrid Mapping c 241
1

1 − Pr(X 1 =1,X 2 = 1)+Pr(X 1 =0,X 2 =0) c
r =1 − . (11.10)
2
Once r is known, θ is determined from the first equation in (11.9) as
1
1 − r − [Pr(X 1 =0,X 2 = 0)] c
θ = . (11.11)
r(1 − r)
Thus, the map (θ, r) → (Pr(X 1 =1,X 2 =1), Pr(X 1 =0,X 2 = 0)) is one
to one. Its range is not the entire set {(s, t): s ≥ 0,t ≥ 0,s + t ≤ 1} since
one can demonstrate that any image point of the map must in addition
satisfy the inequality

Pr(X 1 =1,X 2 =0) 2
≤ Pr(X 1 =1,X 2 =1) Pr(X 1 =0,X 2 =0). (11.12)
See Problem 8 for elaboration.
The observed values of Pr(X 1 =1,X 2 = 1) and Pr(X 1 =0,X 2 =0)
are maximum likelihood estimates for the simplified model in which the
only constraints on the four probabilities displayed in (11.9) are nonnega-
tivity, the symmetry condition Pr(X 1 =1,X 2 =0)=Pr(X 1 =0,X 2 = 1),
and the requirement that the four probabilities sum to 1. This simplified
model has in effect two parameters, which we can identify with the prob-
abilities Pr(X 1 =1,X 2 = 1) and Pr(X 1 =0,X 2 = 0) and estimate by
their empirical values. These values are maximum likelihood estimates un-
der the simplified model. If these estimates satisfy inequality (11.12), then
they furnish maximum likelihood estimates of the radiation hybrid model
as well. Since maximum likelihood estimates are preserved under repara-
meterization, the maximum likelihood estimates of r and θ can then be
computed by substituting estimated values for theoretical values in (11.10)
and (11.11).
Under the polyploid model with many loci, likelihood calculation is hin-
dered by the fact that likelihoods no longer factor. Nonetheless, it is possible
to design a fast algorithm for likelihood calculation based on the theory of
hidden Markov chains [18]. In the current context, there exists a Markov
chain whose current state is the number of markers present in a clone at the
current locus. As the chain progresses from one locus to the next, starting
at the leftmost locus and ending at the rightmost locus, it counts the num-
ber of markers at each locus in the clone. These numbers are hidden from