Information as Maximum-Caliber Deviation: A bridge between Integrated Information Theory and the Free Energy Principle

Information can be defined as the deviation of realized dynamics from a constrained maximum-caliber path ensemble over a finite time horizon. Under this definition, the cause/effect repertoires central to Integrated Information Theory 3.0 emerge directly from maximum-caliber variational principles, allowing IIT's phenomenological calculus to be re-derived from constrained entropy-maximization. This framework provides a theoretical bridge to active inference, which is mathematically dual to this approach under Langevin dynamics. When applied via the Central Limit Theorem for Markov chains and large deviations theory to Ising models, this information measure is equivalent to prediction error under accompanying predictive coding models, potentially explaining the hill-shaped trajectory of Φ observed in neuronal cultures adapting to sensory inputs.

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Abstract

The Free Energy Principle (FEP) is a leading framework for mathematically modeling self-organization and learning, while Integrated Information Theory (IIT) is a computational ontology of consciousness oriented around irreducible cause and effect. While conceptual unifications have been proposed and appear to be supported by empirical findings, the absence of a rigorous mathematical mapping places upper bounds on their precision and testability. This work proposes that information can be defined as the deviation $ψ$ of realized dynamics from a constrained maximum-caliber (MaxCal) path ensemble over a finite time horizon. Under this definition, each of the cause/effect repertoires central to IIT 3.0 emerge directly from MaxCal variational principles, allowing IIT's phenomenological calculus to be re-derived from constrained entropy-maximization (CMEP). This framework supplies a theoretical bridge to active inference, which is mathematically dual to CMEP under Langevin dynamics, and offers a principled route for extending IIT to new dynamical regimes. When the approach is applied under the Central Limit Theorem (CLT) for Markov chains and via large deviations theory (LDT) to Ising models, information $ψ$ is shown to be equivalent to prediction error under accompanying predictive coding models. This may hold relevance to the ``hill-shaped trajectory'' of $Φ$ observed in neuronal cultures adapting to sensory inputs. Together, these results provide a physically and mathematically grounded rationale for studying the convergence of FEP, IIT, and thermodynamic frameworks of cognition such as recent work grounding consciousness in violations of the Fluctuation-Dissipation Theorem (FDT).