General Bayesian theories and the emergence of the exclusivity principle

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.