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Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

David Schmid, John H. Selby, Robert W. Spekkens

TL;DR

This work introduces causal-inferential theories to disentangle causal mechanisms from epistemic inferences within a unified, process-theoretic framework. It defines operational and classical realist CI theories, formalizes inferential equivalence, and shows how quotienting leads to novel GPT-like structures while clarifying the origins of Bell and noncontextuality no-go theorems. The paper proposes Leibnizianity as a rehabilitated, central principle linking inferential equivalence to realist representations and outlines a program for nonclassical realist theories that could reconcile quantum predictions with locality and contextuality constraints. Collectively, the framework provides a structured path toward realistic, quantum-compatible explanations of operational phenomena and offers new avenues for reconstructing quantum theory from foundational axioms.

Abstract

Using a process-theoretic formalism, we introduce the notion of a causal-inferential theory: a triple consisting of a theory of causal influences, a theory of inferences (of both the Boolean and Bayesian varieties), and a specification of how these interact. Recasting the notions of operational and realist theories in this mold clarifies what a realist account of an experiment offers beyond an operational account. It also yields a novel characterization of the assumptions and implications of standard no-go theorems for realist representations of operational quantum theory, namely, those based on Bell's notion of locality and those based on generalized noncontextuality. Moreover, our process-theoretic characterization of generalised noncontextuality is shown to be implied by an even more natural principle which we term Leibnizianity. Most strikingly, our framework offers a way forward in a research program that seeks to circumvent these no-go results. Specifically, we argue that if one can identify axioms for a realist causal-inferential theory such that the notions of causation and inference can differ from their conventional (classical) interpretations, then one has the means of defining an intrinsically quantum notion of realism, and thereby a realist representation of operational quantum theory that salvages the spirit of locality and of noncontextuality.

Unscrambling the omelette of causation and inference: The framework of causal-inferential theories

TL;DR

This work introduces causal-inferential theories to disentangle causal mechanisms from epistemic inferences within a unified, process-theoretic framework. It defines operational and classical realist CI theories, formalizes inferential equivalence, and shows how quotienting leads to novel GPT-like structures while clarifying the origins of Bell and noncontextuality no-go theorems. The paper proposes Leibnizianity as a rehabilitated, central principle linking inferential equivalence to realist representations and outlines a program for nonclassical realist theories that could reconcile quantum predictions with locality and contextuality constraints. Collectively, the framework provides a structured path toward realistic, quantum-compatible explanations of operational phenomena and offers new avenues for reconstructing quantum theory from foundational axioms.

Abstract

Using a process-theoretic formalism, we introduce the notion of a causal-inferential theory: a triple consisting of a theory of causal influences, a theory of inferences (of both the Boolean and Bayesian varieties), and a specification of how these interact. Recasting the notions of operational and realist theories in this mold clarifies what a realist account of an experiment offers beyond an operational account. It also yields a novel characterization of the assumptions and implications of standard no-go theorems for realist representations of operational quantum theory, namely, those based on Bell's notion of locality and those based on generalized noncontextuality. Moreover, our process-theoretic characterization of generalised noncontextuality is shown to be implied by an even more natural principle which we term Leibnizianity. Most strikingly, our framework offers a way forward in a research program that seeks to circumvent these no-go results. Specifically, we argue that if one can identify axioms for a realist causal-inferential theory such that the notions of causation and inference can differ from their conventional (classical) interpretations, then one has the means of defining an intrinsically quantum notion of realism, and thereby a realist representation of operational quantum theory that salvages the spirit of locality and of noncontextuality.

Paper Structure

This paper contains 64 sections, 19 theorems, 322 equations.

Table of Contents

  1. Introduction
  2. Process theories and diagram-preserving maps
  3. Causal primitives
  4. Process theories of laboratory procedures
  5. Process theory of classical functional dynamics
  6. Inferential primitives
  7. Bayesian probability theory
  8. Boolean propositional logic
  9. The full inferential process theory
  10. Causal-inferential theories
  11. Operational causal-inferential theories
  12. Properties of the prediction map
  13. Quantum theory as an operational CI theory
  14. Classical realist causal-inferential theories
  15. Inferential equivalence
  16. ...and 49 more sections

Key Result

Theorem 4.2

For every process $\mathcal{D}\in\textsc{P\!-\!S}$ in the domain of $\mathbf{p}:\textsc{P\!-\!S}\to\textsc{SubStoch}$, i.e., which is causally closed, the image $\mathbf{p}(\mathcal{D})$ of $\mathcal{D}$ under $\mathbf{p}$ is fully specified by the probabilities assigned to atomic propositions on it

Theorems & Definitions (40)

  • Definition 1.1: Processes
  • Definition 1.2: Diagrams
  • Definition 1.3: Process theories
  • Definition 1.5
  • Definition 1.7: Diagram-preserving maps
  • Definition 1.9: Partial diagram-preserving maps
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Definition 5.1: Inferential equivalence for operational CI theories
  • ...and 30 more