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

arXiv: https://arxiv.org/abs/2605.12536 · HTML: https://mlsanity.com/paper/2605.12536/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 26 of its 422 references that are in the map.
- First posted on arXiv 3 May 2026.

## Builds on (most cited first, 26 of 26)

- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Learning deep representations by mutual information estimation and maximization](https://mlsanity.com/paper/1808.06670/) (2018)
- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Exploring the Limits of Language Modeling](https://mlsanity.com/paper/1602.02410/) (2016)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Yes, but Did It Work?: Evaluating Variational Inference](https://mlsanity.com/paper/1802.02538/) (2018)
- [Deep Active Inference as Variational Policy Gradients](https://mlsanity.com/paper/1907.03876/) (2019)
- [Where is the Information in a Deep Neural Network?](https://mlsanity.com/paper/1905.12213/) (2019)
- [Variational Boosting: Iteratively Refining Posterior Approximations](https://mlsanity.com/paper/1611.06585/) (2016)
- [The edge of chaos: quantum field theory and deep neural networks](https://mlsanity.com/paper/2109.13247/) (2021)
- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Posterior and variational inference for deep neural networks with heavy-tailed weights](https://mlsanity.com/paper/2406.03369/) (2024)
- [Measuring Integrated Information: Comparison of Candidate Measures in Theory and Simulation](https://mlsanity.com/paper/1806.09373/) (2018)
- [Online Variational Sequential Monte Carlo](https://mlsanity.com/paper/2312.12616/) (2023)
- [Beyond integrated information: A taxonomy of information dynamics phenomena](https://mlsanity.com/paper/1909.02297/) (2019)
- [The Closure of the Physical, Consciousness and Scientific Practice](https://mlsanity.com/paper/2110.03518/) (2021)
- [Unbiased Active Inference for Classical Control](https://mlsanity.com/paper/2207.13409/) (2022)
- [Free Energy Projective Simulation (FEPS): Active inference with interpretability](https://mlsanity.com/paper/2411.14991/) (2024)
- [Integrated Information Theory: A Consciousness-First Approach to What Exists](https://mlsanity.com/paper/2510.25998/) (2025)
- [A Non-equilibrium Thermodynamic Framework of Consciousness](https://mlsanity.com/paper/2005.02801/) (2020)
- [Only what exists can cause: An intrinsic powers view of free will](https://mlsanity.com/paper/2206.02069/) (2022)
- [On the linear scaling of entropy vs. energy in human brain activity, the Hagedorn temperature and the Zipf law](https://mlsanity.com/paper/2407.13196/) (2024)
- [Understanding the Trade-offs in Accuracy and Uncertainty Quantification: Architecture and Inference Choices in Bayesian Neural Networks](https://mlsanity.com/paper/2503.11808/) (2025)
- [Integrated information and predictive processing theories of consciousness: An adversarial collaborative review](https://mlsanity.com/paper/2509.00555/) (2025)

## Abstract (arXiv)

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 psi 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 psi is shown to be equivalent to prediction error under accompanying predictive coding models. This may hold relevance to the ``hill-shaped trajectory'' of Phi 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).