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Reversing information flow: retrodiction in semicartesian categories

Arthur J. Parzygnat

TL;DR

This work develops a rigorous, process-theoretic framework for retrodiction in semicartesian categories, unifying classical and quantum information flow under a single abstract approach. It extends Bayesian inference and Jeffrey's probability kinematics to arbitrary semicartesian categories by constructing a retrodiction structure that interacts coherently with state-preserving channels and dilations. In the classical case, Bayesian inversion defines a retrodiction functor, while in the quantum setting a Petz-based retrodiction map $\mathscr{R}^{\mathrm{P}}$ yields a retrodiction functor on the category of all states for finite-dimensional $C^{*}$-algebras, with explicit compositionality and involutivity properties. The paper also clarifies the limitations (nonexistence of a universal retrodiction on the full classical/quantum categories) and provides a concrete abstract formulation of probability kinematics beyond standard Bayes’ rule, enabling a broader, theory-driven understanding of backwards belief propagation.

Abstract

In statistical inference, retrodiction is the act of inferring potential causes in the past based on knowledge of the effects in the present and the dynamics leading to the present. Retrodiction is applicable even when the dynamics is not reversible, and it agrees with the reverse dynamics when it exists, so that retrodiction may be viewed as an extension of inversion, i.e., time-reversal. Recently, an axiomatic definition of retrodiction has been made in a way that is applicable to both classical and quantum probability using ideas from category theory. Almost simultaneously, a framework for information flow in in terms of semicartesian categories has been proposed in the setting of categorical probability theory. Here, we formulate a general definition of retrodiction to add to the information flow axioms in semicartesian categories, thus providing an abstract framework for retrodiction beyond classical and quantum probability theory. More precisely, we extend Bayesian inference, and more generally Jeffrey's probability kinematics, to arbitrary semicartesian categories.

Reversing information flow: retrodiction in semicartesian categories

TL;DR

This work develops a rigorous, process-theoretic framework for retrodiction in semicartesian categories, unifying classical and quantum information flow under a single abstract approach. It extends Bayesian inference and Jeffrey's probability kinematics to arbitrary semicartesian categories by constructing a retrodiction structure that interacts coherently with state-preserving channels and dilations. In the classical case, Bayesian inversion defines a retrodiction functor, while in the quantum setting a Petz-based retrodiction map yields a retrodiction functor on the category of all states for finite-dimensional -algebras, with explicit compositionality and involutivity properties. The paper also clarifies the limitations (nonexistence of a universal retrodiction on the full classical/quantum categories) and provides a concrete abstract formulation of probability kinematics beyond standard Bayes’ rule, enabling a broader, theory-driven understanding of backwards belief propagation.

Abstract

In statistical inference, retrodiction is the act of inferring potential causes in the past based on knowledge of the effects in the present and the dynamics leading to the present. Retrodiction is applicable even when the dynamics is not reversible, and it agrees with the reverse dynamics when it exists, so that retrodiction may be viewed as an extension of inversion, i.e., time-reversal. Recently, an axiomatic definition of retrodiction has been made in a way that is applicable to both classical and quantum probability using ideas from category theory. Almost simultaneously, a framework for information flow in in terms of semicartesian categories has been proposed in the setting of categorical probability theory. Here, we formulate a general definition of retrodiction to add to the information flow axioms in semicartesian categories, thus providing an abstract framework for retrodiction beyond classical and quantum probability theory. More precisely, we extend Bayesian inference, and more generally Jeffrey's probability kinematics, to arbitrary semicartesian categories.
Paper Structure (4 sections, 9 theorems, 40 equations, 1 table)

This paper contains 4 sections, 9 theorems, 40 equations, 1 table.

Table of Contents

  1. Introduction and main results
  2. Bayesian retrodiction in classical probability
  3. Semicartesian categories and state-preserving channels
  4. Categorical retrodiction and probability kinematics

Key Result

Theorem 2.11

The assignment $\mathscr{S}^{+}(\mathbf{FinStoch})\xrightarrow{\mathscr{R}^{\mathrm{B}}}\mathscr{S}^{+}(\mathbf{FinStoch})^{\mathrm{op}}$ sending a morphism to its Bayesian inverse satisfies the following properties: In other words, $\mathscr{S}^{+}(\mathbf{FinStoch})$ equipped with $\mathscr{R}^{\mathrm{B}}$ is a symmetric monoidal inverting dagger category Ka18PaBu22.

Theorems & Definitions (53)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Remark 2.9
  • Theorem 2.11
  • Remark 2.12
  • ...and 43 more