Beware of the DAG!
Philip Dawid, University of Cambridge, UK
Conditional independence (CI) is a basic property of a probability distribution. It has a number of important general properties, which allow it to be manipulated in a formal manner. One popular way of doing this is to create a graphical representation: for example (but not the only one) a directed acyclic graph (DAG). There is a clear formal semantics by means of which we can interrogate a DAG to determine just what CI properties of a distribution it represents.
But there are many differences between the properties of probability distributions and those of DAGs. For example, probabilistic CI is a symmetric relationship, whereas the directionality of arrows in a DAG appears to embody a non-symmetric relationship. Although such features of the DAG contribute only in very indirectly to its probabilistic interpretation, there is an almost irresistible temptation to read more into them than the formal semantics require: in particular, to interpret the arrows as having a causal interpretation. Such "reification" of entirely incidental ingredients of the model underlies the enterprise of causal discovery. How, when, and to what extent can it be justified?