= 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.<Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others>[Definitions 17, 19 and 27–28]{ref}
A concrete equality can arise from the positive joint distribution
$$
p(u,a,b,c,d)=p(u)p(a)p(b\mid a,u)p(c\mid a,b)p(d\mid c,u),
$$
with $U$ unobserved. Define
$$
q(d\mid a,c)=\sum_b p(d\mid a,b,c)p(b\mid a).
$$
Substitution 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.<Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others>[Introduction, equation 2 and Figures 1 and 3]{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>.<Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others>[Theorem 46 and Example 47]{ref}
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