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Manfred comments on Deriving probabilities from causal diagrams - Less Wrong Discussion
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I'm pretty sure the solution here is just to assume that the usual iterative procedure is correct. If something can be proven equivalent to that, it works. Even if the d-separation independence thing is just a patch, the correct solution probably won't need many patches because the iterative procedure is simple.
Problem 2
If I find a causal prior and then make observations, my updates can change my probabilities for various parent nodes. E.g. in the marble game, if I condition on a white marble, my probability of Heads changes. But shouldn't conditioning be equivalent to just adding the information you're conditioning on to your pool of information, and rederiving from scratch? And yet if we follow the procedure above, the parent node's probability is totally fixed. What gives?
This actually works if you condition every probability, including the probability of the parent nodes, on the observed information. For example, say that option one is you could start with all options possible in the marble game and then observe that the result was not Heads and White. And option two is you could determine the marble color causally, in a way that never even has the possibility of White when Heads. And these two options result in different probabilities.
This really reinforces how the information about how a node's value is causally generated is different from observed information about that node.