In an article published in 2001, Pearl wrote:
I [Pearl] turned Bayesian in 1971, as soon as I began reading Savage’s monograph The Foundations of Statistical Inference [Savage, 1962]. The arguments were unassailable: (i) It is plain silly to ignore what we know, (ii) It is natural and useful to cast what we know in the language of probabilities, and (iii) If our subjective probabilities are erroneous, their impact will get washed out in due time, as the number of observations increases.
Thirty years later, I [Pearl] am still a devout Bayesian in the sense of (i), but I now doubt the wisdom of (ii) and I know that, in general, (iii) is false.
The bulk of human knowledge is organized around causal, not probabilistic relationships, and the grammar of probability calculus is insufficient for capturing those relationships. Specifically, the building blocks of our scientific and everyday knowledge are elementary facts such as “mud does not cause rain” and “symptoms do not cause disease” and those facts, strangely enough, cannot be expressed in the vocabulary of probability calculus. It is for this reason that I consider myself only a half-Bayesian.
Interesting. The Neyman-Rubin framework of potential outcomes does allow for casual reasoning within a probabilistic structure, but indeed it does not allow for statements such as “mud does not cause rain.” In the potential outcomes notation, one could define a random variable y=1 for rain or 0 for no rain, and define y^1 to be the outcome under treatment and y^2 to be the outcome under control. But it would not make sense for “mud” to be a treatment: in the potential-outcomes framework, a treatment is something that you do, not something such as “mud” that you observe.
I’m not saying here that Pearl’s framework is a good or bad idea; my point here is that I’m agreeing that he indeed seems to be asking questions that cannot be addressed by probability models.
Some of my earlier discussions with Pearl are here.