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:

(U, V) \to A, C; (P, Q) \to B; (R, S) \to D; (A, B) \to F; (C, D) \to M; (F, M) \to X, Y .

Let

f (g)

t (g ∣ h, k)

p (g) = Z \in {U, V, P, Q, R, S} \prod f (g_{Z}) t (g_{A} ∣ g_{U}, g_{V}) t (g_{C} ∣ g_{U}, g_{V}) \times t (g_{B} ∣ g_{P}, g_{Q}) t (g_{D} ∣ g_{R}, g_{S}) t (g_{F} ∣ g_{A}, g_{B}) t (g_{M} ∣ g_{C}, g_{D}) \times t (g_{X} ∣ g_{F}, g_{M}) t (g_{Y} ∣ g_{F}, g_{M}) .

For a biallelic locus with allele-1 frequency

p

f (11) = p^{2}

f (12) = 2 p (1 - p)

and

f (22) = (1 - p)^{2}

. The kernel

t

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.

Every 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.