Synthese 186 (3):651 - 696 (2012)

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 density operators 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. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropie quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of 'conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category
Keywords Bayesian inference  Graphical calculus  Conditional density operator  Symmetric monoidal category  Frobenius algebra
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DOI 10.1007/s11229-011-9917-5
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References found in this work BETA

Causality: Models, Reasoning and Inference.Judea Pearl - 2000 - Cambridge University Press.
Probability, Frequency, and Reasonable Expectation.Richard T. Cox - 1946 - American Journal of Physics 14 (2):1-13.
Probability, Frequency and Reasonable Expectation.Richard T. Cox - 1946 - Journal of Symbolic Logic 37 (2):398-399.
Quantum Quandaries: A Category-Theoretic Perspective.J. C. Baez - 2006 - In Dean Rickles, Steven French & Juha T. Saatsi (eds.), The Structural Foundations of Quantum Gravity. Clarendon Press.

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