= Fixing operation
{title2=$\phi_v(q;G)$}
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.<Nested Markov Properties for Acyclic Directed Mixed Graphs by Thomas S. Richardson and others>[Definitions 17 and 19]{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 $B$ gives
$$
\phi_B(p)(a,c\mid b)=\frac{p(a,b,c)}{p(b\mid a)}=p(a)p(c\mid b).
$$
Ordinary 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.
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