Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 45 / 1 / a

Treat $G_i$ as the entire phase-known genotype of individual $i$ at the genetic loci being studied, and let $\overline{F}$ denote the nonfounders. Assume unrelated pedigree founders have independent genotype priors, each nonfounder's genotype depends only on its parents' genotypes, and the observed phenotypes are conditionally independent given the genotypes. These are the standard assumptions behind the pedigree likelihood; shared environmental effects or related pedigree founders would require additional factors.

The directed ancestry graph is acyclic, even when there is inbreeding. Order individuals with parents preceding offspring and apply the chain rule for probabilities. The parental conditional independence assumption givesConditional phenotype independence gives $P(X_1,\dots ,X_n\mid G_1,\dots ,G_n)={\prod }_{i}P(X_i\mid G_i)$. Multiplying these two expressions and summing over every possible latent genotype configuration provesThis is the law of total probability, not a claim that relatives have independent unconditional genotypes.