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]

The fixing operation in a nested Markov model turns a random vertex of a conditional acyclic directed mixed graph into a fixed input and transforms its probability kernel. A vertex $v$ is fixable if no other vertex in its bidirected-connected district of an acyclic directed mixed graph is a descendant of $v$. Graphically, fixing removes edges with an arrowhead at $v$ and moves $v$ from the random to the fixed vertex set.^[ref]

For a positive Markov kernel $q$, the transformation divides $q$ by the conditional factor for $v$ given its district's other vertices and the parents of that district. Fixed arguments are retained as kernel inputs. Fixability is what permits the resulting object to be a normalized kernel over the remaining random variables. Valid fixing sequences must check this condition again after each step.

For the directed acyclic graph $A\to B\to C$, write $p(a,b,c)=p(a)p(b\mid a)p(c\mid b)$. Fixing $B$ givesOrdinary conditioning instead gives $p(a\mid b)p(c\mid b)$. If $A$ is a fair binary variable and $p(B=1\mid A=1)=0.8$, $p(B=1\mid A=0)=0.2$, then conditioning on $B=1$ makes $p(A=1)=0.8$, whereas fixing $B=1$ leaves it $0.5$. The operation is algebraically defined even without interpreting a graph causally.