Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 39 / 3 / d / i / Solution

Label the full siblings at the bottom $X,Y$ and their cousin parents $F,M$. Let $F$ have parents $A,B$ and $M$ parents $C,D$, where $A,C$ are siblings with parents $U,V$. Include the other great-grandparents as $P,Q$, parents of $B$, and $R,S$, parents of $D$. This gives fourteen people, with six independent founders in a pedigree $U,V,P,Q,R,S$.

The pedigree graphical model has the following Directed acyclic graph, with both members of each parent pair pointing to every listed child:Let $f(g)$ be the founder genotype law under Hardy-Weinberg equilibrium, and let $t(g\mid h,k)$ be the Mendelian segregation kernel. The joint probability of all genotypes isFor a biallelic locus with allele-1 frequency $p$, $f(11)=p^2$, $f(12)=2p(1-p)$ and $f(22)=(1-p{)}^2$. The kernel $t$ assigns probability $1/4$ to each labelled pair of parental transmissions and adds these probabilities when they produce the same unordered genotype. Summing this Bayesian network factorization over unobserved genotypes gives their marginal distribution or a pedigree likelihood.