Picturing classical and quantum Bayesian inference

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers mixed quantum states rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. In the case of quantum-like calculi, the latter will be non-commutative. We identify a graphical property that characterizes classical Bayesian inference. The abstract classical Bayesian graphical calculi also allow to model relations among classical entropies, and reason about these. We generalize conditional independence to this very general setting.