 Nested Markov model

A nested Markov model is a statistical model associated with an acyclic directed mixed graph. Besides ordinary conditional independence, it imposes Markov restrictions on probability kernels obtained through valid sequences of a fixing operation. Fixing changes a random vertex into a fixed input and transforms the distribution by an appropriate conditional-density ratio. It is not simply conditioning. Formally, every kernel reached by a valid sequence must satisfy the global Markov property of its reached graph.^[ref]

A concrete equality can arise from the positive joint distributionwith $U$ unobserved. DefineSubstitution and Bayes' theorem reduce this to ${\sum }_{u}p(d\mid c,u)p(u)$, independent of $a$. This is a Verma constraint; the graph can impose it without implying any ordinary conditional independence among the observed variables.^[ref]

For a numerical instance, make $U,A$ independent fair binary variables, and choose success probabilities $p(B=1\mid a,u)=0.1+0.2a+0.4u$, $p(C=1\mid a,b)=0.15+0.2a+0.3b$, and $p(D=1\mid c,u)=0.1+0.3u+0.4c$. Then $q(D=1\mid a,c)=0.25+0.4c$ for either $a$. Yet $p(D=1\mid A=0,C=0)=181/760$ differs from $p(D=1\mid A=1,C=0)=29/125$.

A latent-variable model can impose additional inequalities, so its observed margin may be strictly smaller than the associated nested Markov model.^[ref]