Question to author:
In the part of Chapter 1 that you kindly sent me, a functional, causal model is clearly defined by a set of equations in (1.40). The set provides a joint probability distribution of the variables using a specific order. That distribution may be manipulated to obtain an equivalent probability specification in any other order. I showed in my note that this probability structure could be described by a set of equations in an order different from that of (1.40). (That proof may be wrong, though on p. 31 you suggest the result was known in '93.) Consequently (1.40) can be replaced by a different set of equations. You tell us now to see what happens were a variable to be controlled; this in terms of the set, and I showed that different consequences flowed if different sets were used. How do I decide which set is correct?
To scientists who grew up in the age of empiricism (e.g., you and me), the question:
"How do I decide which set is correct?"
often amounts to asking
"How do I decide, by looking at the available data, which set is correct?"
The answer to this latter question is, of course, "impossible!"; if the data were capable of helping us decide, then the functional model would not add any information to what we already have in the distribution, but distributional information, as we all know, is insufficient for answering any decision-related query.
So, the information as to which set of equations is appropriate must come from a different source, not from the data. The most reliable source of this information (and one that is most acceptable to empiricists) comes from data obtained in the past, under various experimental conditions, including randomized trials. For example, if we (or other scientists) perform controlled experiments and find that wetting the pavement does not make the rain fall, it gives us the lisenced to write the equation
Note that the functional form of f(*) need not be specified in those models (unless we ask counterfactual questions), only the set of arguments counts.
Next discussion (Lindley (2): On causality and decision trees)