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A Gaussian Bayesian network specifies a linear normal conditional model at each node of a Bayesian network: $X_j\mid X_{\mathrm{pa}(j)}\sim N({\alpha }_j+{\gamma }_j^T{X}_{\mathrm{pa}(j)},{\sigma }_j^2)$. Independent local errors and an acyclic ordering generate a joint multivariate normal distribution. Positive conditional variances give a nonsingular joint multivariate normal distribution. The graph constrains regression coefficients and hence conditional independence.

The junction tree algorithm performs exact inference in a finite probabilistic graphical model. First moralize a Bayesian network if necessary; then apply triangulation of an undirected graph, organize its maximal cliques into a junction tree, and assign each original factor to one containing clique. Inward and outward junction-tree sum-product messages compute the normalizing constant and marginal distributions. Its cost depends exponentially on the largest clique's number of variables, rather than necessarily on the total number of variables.

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.

The Bayesian network has a shared parent $\psi$ and, for each hospital, the chain $\psi \to {\theta }_i\to y_i$. A plate encloses the hospital-specific pair. Observed counts are shaded; the local rates are latent variables. Conditional on $\psi$, the pairs are independent.

The original PDF shows the chain $A\to C\to B$, not a collider. Its Bayesian network factorization is $p(a,b,c)=p(a)p(c\mid a)p(b\mid c)$. For any $c$ with positive marginal probability,Integrating or summing over $a$ also gives the same conditional marginal $p(b\mid c)$. ThusThis is conditional independence; the graph does not generally imply marginal independence, since summing $p(a)p(c\mid a)p(b\mid c)$ over $c$ can transmit dependence from $A$ to $B$. Special statistical parameter choices can make marginal independence hold as well, so the claim is that only the conditional statement is guaranteed by the graph. Conditional distributions on null values of $C$ are immaterial to this assertion.

Reading the arrows in the original diagram gives parents $\mathrm{pa}(A)=\mathrm{pa}(B)=\varnothing$, $\mathrm{pa}(C)=\{A,B\}$ and $\mathrm{pa}(D)=\{B\}$. The Bayesian network factorization isFor binary variables, each conditional probability table has one free probability per parent configuration, since its two entries sum to one. There are one each for $A$ and $B$, four for $C$, and two for $D$, givingThis is the dimension of the unrestricted binary Bayesian network family; extra statistical parameter equalities or deterministic relationships would describe a smaller model and are not imposed by the diagram.

Assign one genotype random variable to each person in a pedigree. Independent founders in a pedigree carry population genotype laws, and each other genotype has its conditional distribution given its two parents from Mendelian segregation. The ancestry Directed acyclic graph is a Bayesian network for the joint genotype law. Its moral graph contains each parent-child edge and each coparent edge. Each founder factor and each child-with-parents factor belongs to a clique, giving an undirected graphical representation. The graph is an independence representation; equality with the minimal conditional independence graph needs additional faithfulness assumptions, particularly because inheritance can produce zero probabilities.