 Solution

The pedigree founder factor $P(G_i)$ depends on population allele and haplotype frequencies, ancestry, mating assumptions and any population linkage disequilibrium. At one autosomal genetic locus, Hardy-Weinberg equilibrium with allele frequencies $p,q$ gives genotype probabilities $p^2,2pq,q^2$. For a specified ordered pair of independent pedigree founder haplotypes $h_1,h_2$, the probability is $p_{h_1}{p}_{h_2}$; an unordered distinct pair has twice that probability. Pedigree founders assumed to be related require a joint prior rather than a product of unrelated-founder factors.

The nonfounder factor depends on the parental genotypes and phases, Mendelian segregation, and the recombination fractions between genetic loci. In a two-locus heterozygote of known phase, the two parental haplotypes have transmission probabilities $(1-\theta )/2$ each, and the two recombinant haplotypes $\theta /2$ each. Multiply the maternal and paternal gamete probabilities and sum any gamete combinations producing the child's genotype. Sex-specific genetic recombination, mutation or segregation distortion can be incorporated by changing these gamete probabilities. At one ordinary autosomal genetic locus, each parental copy is transmitted with probability $1/2$.