Bayesian Conditioning, the Reflection Principle, and Quantum Decoherence
The probabilities a Bayesian agent assigns to a set of events typically change with time, for instance when the agent updates them in the light of new data. In this paper we address the question of how an agent’s probabilities at different times are constrained by Dutch-book coherence. We review and attempt to clarify the argument that, although an agent is not forced by coherence to use the usual Bayesian conditioning rule to update his probabilities, coherence does require the agent’s probabilities to satisfy van Fraassen’s  reflection principle (which entails a related constraint pointed out by Goldstein ). We then exhibit the specialized assumption needed to recover Bayesian conditioning from an analogous reflection-style consideration. Bringing the argument to the context of quantum measurement theory, we show that “quantum decoherence” can be understood in purely personalist terms—quantum decoherence (as supposed in a von Neumann chain) is not a physical process at all, but an application of the reflection principle. From this point of view, the decoherence theory of Zeh, Zurek, and others as a story of quantum measurement has the plot turned exactly backward.
KeywordsProbability Assignment Fair Price Lottery Ticket Dutch Book Ticket Price
We thank Lucien Hardy for persisting that the example in Sect. 15.2 should be important to us. We thank Matthew Leifer for bringing the work of Goldstein  to our attention, which derivatively (and slowly) led us to an appreciation of van Fraassen’s reflection principle ; if we were quicker thinkers, this paper could have been written 6 years ago. This work was supported in part by the U. S. Office of Naval Research (Grant No. N00014-09-1-0247).
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