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In the modern Bayesian view classical probability theory is simply anextension of conventional logic, i.e., a quantitative tool that allows forconsistent reasoning in the presence of uncertainty. Classical theorypresupposes, however, that--at least in principle--the amount of evidence thatan experimenter can accumulate always matches the size of the hypothesis space.I investigate how the framework for consistent reasoning must be modified innon-classical situations where hypotheses form a continuum, yet the maximumevidence accessible through experiment is not allowed to exceed some finiteupper bound. Invoking basic consistency requirements pertaining to thepreparation and composition of systems, as well as to the continuity ofprobabilities, I show that the modified theory must have an internal symmetryisomorphic to the unitary group. It thus appears that the only consistentalgorithm for plausible reasoning about a continuum of hypotheses on the basisof finite evidence is furnished by quantum theory in complex Hilbert space.

Consistent reasoning about a continuum of hypotheses on the basis of finite evidence