Bayesian inference and retrodiction for faithful states on von Neumann algebras

• URL: http://arxiv.org/abs/2608.20001v1
• Date: Thu, 20 Aug 2026 13:17:30 GMT
• Title: Bayesian inference and retrodiction for faithful states on von Neumann algebras
• Abstract summary: Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference.<n>Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory.<n>This paper extends these results to the infinite-dimensional setting on von Neumann algebras.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.

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