  We address the problem of reconstructing quantum theory from the perspective
of an agent who makes bets about the outcomes of possible experiments. We build
a general Bayesian framework that can be used to organize the agent's beliefs
and update them when new information becomes available. Our framework includes
as special cases classical and quantum probability theory, as well as other
forms of probabilistic reasoning that may arise in future physical theories.
Building on this framework, we develop a notion of ideal experiment, which in
quantum theory coincides with the notion of projective measurement. We then
prove that, in every general Bayesian theory, ideal experiments must satisfy
the exclusivity principle, a property of projective measurements that plays a
central role in the characterization of quantum correlations. Our result
suggests that the set of quantum correlations may be completely characterized
in terms of Bayesian consistency conditions.
