I would certainly not quibble with your interpretation of the "do" operator as an important "standard" way of describing certain important kinds of intervention, and there is no doubt that its semantics and application deserve very careful consideration.
Now it may or may not be the case that many (if not all) of
the various manipulations can be reduced to "do". What
certainly is the case is that all applications of "do" are
special forms of a more general manipulation operation.
If we want a broad and fundamental theory of
causal inference, it seems to me inappropriate to base it on a
special case, no matter how important that might be. The
language of decision trees, influence diagrams, conditional
independence manipulations of intervention variables, etc.,
is perfect for describing and manipulating
the requisite degree of additional generality, and, very
importantly, by allowing us to express explicitly what we
mean when we specialize it to
the kind of problems covered by your interpretation of "do," allows us to
speak clearly and without fear of misunderstanding. Rules such as those
in the Theorem 3.4.1 of your book, while perfectly valid (and readily
derivable through influence diagrams) in such special cases, do not appear
to me to be sufficiently fundamental to form the basis of a "calculus".
I was about to elaborate on how science thrives on reductions to basic concepts, how science chooses its basic concepts, why the chosen concepts are necessarily "special cases", and why it is inevitable, therefore, that "broad and fundamental" theories be based on "special cases" (see my reply to Bill Shipley). But then it occurred to me that the (almost perfect) analogy with chemistry would be instructive at this point, and that the following hypothetical letter would convey the message better than any philosophical discussion. Here is a letter that could have been received by John Dalton from one of the many contemporary critics of his atomic theory of matter.
Dear Mr. Dalton,
Now it may or may not be the case that many (if not all) of the various chemical compounds in the universe can be reduced to the chemical elements. What certainly is the case is that the elements are special forms of a more general chemical configuration. If we want a broad and fundamental theory of chemistry, it seems to me inappropriate to base it on a special case, no matter how important that might be.
A Specific Comment:
The rules in Theorem 3.4.1 of Causality are not derivable through influence diagrams because, when the intervention variable Fi is not idle (see Fig. 3.2), the diagram fails to display the absence of connection between Xi and PAi. For example, I do not know of a way to derive Eq. (3.47) directly from the influence diagram. But, as always, I am open to learning new techniques.
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