Before we return to the question of ordering loci along chromosomes, we briefly consider Fisher's 1992 application of maximum likelihood (ML) to the estimation of recombination fractions. He had proposed ML in his first paper in 1912, but it was in 1922 in his famous paper ``On the mathematical foundations of theoretical statistics" that the many virtues of MLEs were first pointed out (consistency, sufficiency, efficiency etc). In that same year, he applied his method to obtaining efficient estimates of several recombination fractions from data on Drosophila (willistoni, not melanogaster).
We follow his initial discussion; the full data involved 8 loci. One of the data sets referred to the traits scute, beaded and rough:
Fisher says ``Within such a small range, double crossing over may be ignored; yet it would be wrong to use such inconsistency as an argument against the linear arrangement of the genes".
He goes on to estimate recombination fractions 
and 
by maximising:
.
In fact he took logs and linearized. This calculation can be described as the first multilocus mapping using ML, although the data are only on pairwise recombination fractions.
Exercise 3: Complete this calculation both exactly and by linearizing.