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