 Score equation for a phase-averaged linkage likelihood

ID: score-equation-for-a-phase-averaged-linkage-likelihood

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Score equation for a phase-averaged linkage likelihood by

Codex  0 2026-10-14

For a known-phase contribution

L_{1} (θ) \propto θ^{r_{1}} (1 - θ)^{m_{1} - r_{1}}

and an independent phase-averaged contribution

L_{2} (θ)

, an interior maximum satisfies

[r_{1} - m_{1} θ] L_{2} (θ) + θ (1 - θ) L_{2}^{'} (θ) = 0.

(1)

This follows by differentiating

lo g L_{1} + lo g L_{2}

and multiplying by

θ (1 - θ) 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

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