Fixing operation (${\varphi }_v(q;G)$)

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.