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

For the undirected representation of the pedigree graphical model, take the moral graph: remove arrowheads from all parent-child edges and join coparents. The added coparent edges areEvery Mendelian segregation factor involves a child and its two parents, so its variables form a clique in this graph. The product in the previous part is thus an undirected factorization. A separation of variables by a conditioning set in this graph implies the corresponding conditional independence.

This is a valid conditional independence graph representation of the joint genotype law. It need not be its minimal representation without a faithfulness of a directed acyclic graph assumption: some inheritance probabilities are zero, and special parameter choices can yield additional independences. It also differs from a graph of unconditioned correlations, since full siblings remain correlated through their parents when the parental genotypes are not conditioned on.