You want the widget problem in chapter 14 of Jaynes’ Probability Theory: The Logic Of Science; it is extremely similar to the problem you present. Long story short: express each of the known probabilities as expectations of indicator variables (e.g. P(A∩B)=E(I(A∩B))), then maximize entropy subject to the constraints given by those expectations. Jaynes covers a bunch of conceptual arguments for why that’s a sensible procedure to follow.
To do better than that, the next step would be to look for any structure/pattern in the known probabilities and exploit those—e.g. if the known probabilities approximately factor over a certain Bayes net, then a natural guess is that the unknown probabilities will too, which may allow backing out of the unknown probabilities.
You want the widget problem in chapter 14 of Jaynes’ Probability Theory: The Logic Of Science; it is extremely similar to the problem you present. Long story short: express each of the known probabilities as expectations of indicator variables (e.g. P(A∩B)=E(I(A∩B))), then maximize entropy subject to the constraints given by those expectations. Jaynes covers a bunch of conceptual arguments for why that’s a sensible procedure to follow.
To do better than that, the next step would be to look for any structure/pattern in the known probabilities and exploit those—e.g. if the known probabilities approximately factor over a certain Bayes net, then a natural guess is that the unknown probabilities will too, which may allow backing out of the unknown probabilities.
Wow, this is exactly what I was looking for! Thank you so much!