Statement to all:
Apart from the innate symmetricity of structural equation systems, the very definition for the conditional probability as well as Bayes's law of inverse probability unambiguously suggest the reversibility of cause-effect relationships. ...
``A is a cause of B only when B can cause A.'' Amen.
(Quoted from Mr. Asha review of Causality, amazon.com, August 29, 2000)
It is instructive to compare the apparent symmetries of Bayes rule and structural equations with the asymmetry inherent in out conception of causation. Bayes rule asserts that if A is relevant to B, then B must be relevant to A, that is, if learning A changes the probability of B then learning B must change the probability of A. This symmetry constitutes the first of the five properties of graphoids (Causality, page 11). However, this symmetry refers to informational relevance, not to causal relevance (as axiomatized in Causality, Section 7.3.3). Bayes's rule and information relevance deal with changes in one's beliefs about a static world; causality deals with changes in the world itself. My belief about the rain may change in light of reading the barometer, but there is nothing I can do to the barometer that will change the rain in the physical world.
The apparent symmetry of structural equations is equally deceiving. On page 160 I explain how structural equations serve a dual purpose, observational and interventional, and I argue that it is the inteventional component which distingushes structural equations from algebraic equations. Referring to the equation y=bx+e, page 160 reads:
"Note that the operational reading just given makes no claim
about how X (or any other variable) will behave when we
control Y. This asymmetry makes the equality signs in
structural equations different from algebraic equality signs;
the former act symmetrically in relating observations on X and Y
(e.g., observing Y=0 implies bx= -e),
but they act asymmetrically when it comes to interventions
(e.g., setting Y to zero tells us nothing about the relation
between x and e). The arrows in path diagrams
make this dual role explicit, and this may account for the
insight and inferential power gained through the use of diagrams.
Although the literature on structural equation models does not explicitly acknowledge this basic interpretation of structural equations -- a puzzling phenomenon that I explain in Section 5.1 -- it is implicit in the conclusions that scientists draw from SEM studies.
Next discussion (many readers: Errata list for the 3rd printing of Causality)