Pedigree likelihood

For latent genotypes $G_i$ and observed phenotypes $X_i$, assume conditional phenotype independence, independent founder genotypes, and conditionally independent Mendelian transmissions. The likelihood function isThis follows by the chain rule applied to the acyclic ancestry graph, then the law of total probability over all latent genotypes. Penetrance supplies the phenotype factors and genetic recombination supplies multilocus transmission factors.

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

Assign one genotype random variable to each person in a pedigree. Independent founders in a pedigree carry population genotype laws, and each other genotype has its conditional distribution given its two parents from Mendelian segregation. The ancestry Directed acyclic graph is a Bayesian network for the joint genotype law. Its moral graph contains each parent-child edge and each coparent edge. Each founder factor and each child-with-parents factor belongs to a clique, giving an undirected graphical representation. The graph is an independence representation; equality with the minimal conditional independence graph needs additional faithfulness assumptions, particularly because inheritance can produce zero probabilities.

Pedigree peeling eliminates latent genotypes by moving sums inside products whenever the other factors do not depend on the eliminated genotype. For one parent genotype $g$ and one child genotype $h$, the likelihood becomesThe inner bracket is the child's message to its parent. With two states each, it needs four multiplications and two additions; multiplying the two parent weights and messages needs four more multiplications and the final sum one addition. The unrearranged four terms instead use twelve multiplications. Larger pedigrees use the same variable-elimination identity, with additional handling of pedigree loops.