Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 29 / 6 / iv / Solution

Let $s$ index compatible parental phases in family 2, with fixed prior conditional weights ${\pi }_s$. For a given phase, let $r_s$ be the recombinant count and $m_s$ the informative-transmission count, absorbing any fixed genotype factors into $C_s$. The appropriate pedigree likelihood isThis is phase averaging in a linkage likelihood; maximizing over a phase after observing the offspring would not be the same likelihood. In the common special case of two equally likely phases that exchange recombinant and nonrecombinant labels across $m_2$ transmissions, it reduces, up to a constant, toThe constrained maximum-likelihood estimate compares all stationary points and endpoints on $[0,1/2]$.

The actual phase set, weights, counts and numerical maximum for family 2 cannot be determined without its missing pedigree. The special two-phase expression is a conditional example, not an assertion about this unidentified family.