Phase averaging in a linkage likelihood ($L(\theta )={\sum }_s{\pi }_s{L}_s(\theta )$)

An unphased multilocus genotype can have several pairs of haplotypes. Its pedigree likelihood must average over these possible phases with prior weights ${\pi }_s$, rather than maximize over phases after observing the offspring. Under linkage disequilibrium, the weights need not be equal. They can be obtained by conditioning population haplotype frequencies on the observed parental genotype; different choices of what is conditioned on change only parameter-independent factors when those frequencies are held fixed.

For a known-phase contribution $L_1(\theta )\propto {\theta }^{r_1}(1-\theta {)}^{m_1-r_1}$ and an independent phase-averaged contribution $L_2(\theta )$, an interior maximum satisfiesThis follows by differentiating $\log L_1+\log L_2$ and multiplying by $\theta (1-\theta )L_2$. The constrained maximum-likelihood estimate on $[0,1/2]$ must also be compared with the endpoints. An interior root cannot be asserted without the observed pedigree counts; complete nonrecombinant data, for example, have their maximum at zero.

// Target: biology.bigb