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You have written “The consistent histories approach was initiated in order to deal with probabilities in the quantum context, and I think it has solved the problem–while I allow that there may be alternative or better approaches.”
I think that the consistent description of probabilities in quantum mechanics needs more. Probability approach to quantum mechnaics must at the first step to explain why it is possible that the real physical outputs depend on the “which way information” !! This means that real physics depends on the things like the information !! This is shocking !! I have been trying to do this in arXiv:1008.0295. In this approach I have tried to build the extended probability theory appropriate for the describing quantum mechanics. I hope I have arrived at the point to understand the point of the problém. But this theory is not complete. The details can be found in http://arxiv.org/pdf/1008.0295.pdf.