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Manfred comments on Causation, Probability and Objectivity - Less Wrong
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Hm, good point.
I think the oddness is because good Bayesians usually don't treat the real world as probabilistic, but a pearl-ian model of the world has inherent probabilities. Ideal reasoners can't be automatically right about causal relations, and that's because probability is in two different "places" in the two models. The ideal reasoner is automatically right about its probabilities, but it isn't automatically right about these inherent probabilities in the territory. There's no problem with the probability not corresponding exactly to the world, because that's usually how it is - probabilities are merely the best you can do.
So huh.
I said this once above, but it's worth repeating -- Pearl's view of causality has nothing to do with probabilities. It's a fully deterministic theory which can be augmented by modeling uncertainty via probability theory if you want.
A causal model uniquely specifies a bunch of conditional probabilities, right?
Only in a sense that a first order theory of natural numbers does. Really I think it is more accurate to view "a causal model" as a model in the mathematical logic sense -- an object about which logical assertions can be made. In the case of causal models, these assertions are modelling "interventions." Here's a paper on this:
http://www.jair.org/papers/paper648.html
This view appears in Pearl's chapter 7, as well.