Quantum mechanics can be seen as a set of instructions how to calculate probabilities by associating mathematical objects to physical procedures, like preparation, manipulation, and measurement of a system. Quantum theory then yields probabilities which are neutral with respect to its use, e.g., in a Bayesian or a frequentistic way. We investigate a different approach to quantum theory and phyiscal theories in general, in which we aim for subjective predictions in the Bayesian sense. This gives a structure which is very different from the operational framework of general probabilistic theories.We explore these differences and the according mathematical language of such probabilistic Bayesian theories.