From: Dennis Lindley

Subject: Reciprocal links in structural equations

**Question to author:**

Equations (1.42) and (1.43) and the general issue of
description by equations, still perplex me. It is
incoherent to state both *p*(*x*|*y*) and
*p*(*y*|*x*). (Try it
with *x* and *y* binary, when these statements describe 4
values, whereas we know only 3 are needed for the joint
distribution of *x* and *y*.) There are special cases,
as with normal, linear regression, where the
coherence is avoided. Generally I do not see how there
can be two links between *x* and *y*.

**Author's reply**

The coherence problem you describe is precisely
why we do not find any cycles in the graphical
models used in statistics. We do find them in econometrics though.
Graphical models in statistics are defined by
conditional probability claims, while structural equations
in general say nothing about *p*(*x*|*y*) or
*p*(*y*|*x*).
(See footnote 4, page 137). The only claims these
equations make are about *p*(*x*|*do*(*y*)) and *p*(*y*|*do*(*x*))
Moreover, unlike *p*(*x*|*y*) and *p*(*y*|*x*), there are no
coherence constraints on *p*(*x*|*do*(*y*)) and *p*(*y*|*do*(*x*)).
The only coherence constraints we need are:

Next discussion (Asha: *The impossibility of
asymmetric causation*)