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We address the problem of reconstructing quantum theory from the perspectiveof an agent who makes bets about the outcomes of possible experiments. We builda general Bayesian framework that can be used to organize the agent's beliefsand update them when new information becomes available. Our framework includesas special cases classical and quantum probability theory, as well as otherforms of probabilistic reasoning that may arise in future physical theories.Building on this framework, we develop a notion of ideal experiment, which inquantum theory coincides with the notion of projective measurement. We thenprove that, in every general Bayesian theory, ideal experiments must satisfythe exclusivity principle, a property of projective measurements that plays acentral role in the characterization of quantum correlations. Our resultsuggests that the set of quantum correlations may be completely characterizedin terms of Bayesian consistency conditions.

General Bayesian theories and the emergence of the exclusivity principle