Score equation for a phase-averaged linkage likelihood

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