Dear Aurélien,
The consistent histories (CH) approach is best thought of as an interpretation of QM in terms of ‘events’, not just measurement outcomes, inside a closed quantum system, without making any reference to things outside this system. Measurement apparatus, if any, is to be included as part of the closed system, and described in fully quantum terms. Measurements are simply examples of physical processes, and the rules for describing them are the same as for all quantum processes. If you want to look into it, the best place to begin is one of the shorter articles I have written, as indicated in the Consistent Histories Essentials of the IJQF workshop.
Now to your specific questions. The CH approach allows retrodiction from measurement outcomes to previous properties; e.g., in the EPR-Bohm situation, if Alice measures S_x and gets the value -1/2, she is justified in concluding that the particle had this value of spin just before it entered her apparatus. This is worked out in some detail in [1]. (This topic also came up in the IJQF workshop in the Time-symmetric theories section where I started a thread Retrocausation vs Retrodiction.) However, the discussion of events, microscopic or macroscopic, is not limited to situations where there are measurements, so there is not a limitation to actual measurements, as in Bohr’s approach. We consistent historians regard our approach as ‘Copenhagen done right’: we show that the results which emerge from the usual calculations based on textbook presentations are (usually) correct, but the arm waving that accompanies them is often misleading or unnecessary.
The consistency conditions are employed to single out families of histories to which probabilities can be assigned using the extended Born rule, probabilities which are not limited to measurement outcomes. Whereas in a given consistent family only one history will be actualized, CH is a probabilistic interpretation of QM, and standard (Kolmogorov) probability theory requires a sample space of mutually exclusive possibilities. I am not sure these remarks are addressing your concerns, and it might help to take up a specific example if you have one in mind. (E.g., various history families are discussed in [1].)
Regarding Bohm and Hiley. It is twenty years since I looked a their treatment, so I had to pull out my old notes. Their discussion is based entirely on the earliest Gell-Mann and Hartle publication in 1990; there is no reference to my work or that of Omnès. At that time (1990) the CH approach was still undergoing substantial development; my own ideas were not entirely clear until the mid 1990s. So a lot of the Bohm and Hiley discussion is out of date. You will find a list of later criticisms of CH, though not from the BM perspective, in [2].
As for Shelly’s 1998 presentation in Physics Today, it is full of mistakes; unfortunately he did not take the trouble of sending me a copy in advance, so I had to write a lengthy letter published in a later Physics Today; it you want I can try and dig up the reference.
Your attachments were already in my files, but I took another look at your 2006 PLA to refresh my mind. It is too bad you were not aware of my work on Bohmian trajectories, as I think you would have found it much harder to refute.
I hope these comments are of some help, and that things have cooled down a bit in Grenoble. Otherwise you should plan to take your laptop up into the mountains, which I understand are very impressive.
Best wishes! Bob Griffiths
[1] EPR, Bell, and quantum locality. Am. J. Phys., 79:954–965, 2011. arXiv:1007.4281.
[2] R. B. Griffiths, “A Consistent Quantum Ontology”, Stud. Hist. Phil. Mod. Phys. 44 (2013) 93; arXiv:1105.3932
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