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arxiv: 2605.12536 · v1 · submitted 2026-05-03 · 🧬 q-bio.NC · cs.AI· cs.IT· math.IT

Recognition: unknown

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

Alexander Kearney
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:07 UTC · model grok-4.3

classification 🧬 q-bio.NC cs.AIcs.ITmath.IT
keywords integrated information theoryfree energy principlemaximum caliberprediction errorcause-effect repertoiresactive inferencevariational principlesconsciousness modeling
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The pith

Information is the deviation of realized dynamics from a constrained maximum-caliber path ensemble, from which IIT 3.0's cause-effect repertoires emerge via variational principles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proposes defining information as the deviation ψ of actual dynamics from what a constrained maximum-caliber path ensemble would produce over a finite time horizon. Under this definition, the cause-effect repertoires and integrated information measures of IIT 3.0 arise directly from maximum-caliber variational principles, re-deriving the theory's phenomenological calculus from constrained entropy maximization. The same deviation is shown to equal prediction error in predictive coding models for Markov chains under the central limit theorem and for Ising models under large deviations theory. A sympathetic reader would care because the construction supplies a precise mathematical mapping between integrated information theory and the free energy principle's active inference framework, opening routes to extend both to new dynamical systems.

Core claim

The central claim is that information can be defined as the deviation ψ of realized dynamics from a constrained maximum-caliber path ensemble, from which each of the cause/effect repertoires central to IIT 3.0 emerge directly from MaxCal variational principles. This re-derives IIT's phenomenological calculus from constrained entropy-maximization, supplies a theoretical bridge to active inference which is mathematically dual under Langevin dynamics, and shows that ψ equals prediction error under the central limit theorem for Markov chains and large deviations theory for Ising models.

What carries the argument

The deviation ψ of realized dynamics from a constrained maximum-caliber path ensemble, which acts as the definition of information and generates IIT's cause-effect structures from variational entropy maximization.

If this is right

  • IIT 3.0's integrated information measures can be obtained from constrained entropy maximization alone.
  • The information measure ψ is equivalent to prediction error in predictive coding models for Markov chains and Ising models.
  • The framework supplies a principled route for extending IIT to dynamical regimes beyond its current scope.
  • It provides a rationale for studying convergence among FEP, IIT, and thermodynamic accounts of cognition such as fluctuation-dissipation violations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The unification may predict that integrated information follows a hill-shaped trajectory during adaptation to sensory inputs in neural systems.
  • The approach could be tested by comparing ψ values computed from observed trajectories against empirical measures of information integration in biological preparations.
  • Consciousness-related quantities might be re-interpreted as measurable deviations from maximum-entropy path ensembles in physical systems.

Load-bearing premise

The proposed definition of information as maximum-caliber deviation is sufficient to recover the full set of IIT 3.0 cause-effect repertoires and measures without additional unstated constraints.

What would settle it

A direct computation on a small Markov chain or Ising model in which the cause-effect repertoires obtained from the maximum-caliber deviation do not match the repertoires produced by standard IIT 3.0 procedures.

Figures

Figures reproduced from arXiv: 2605.12536 by Alexander Kearney.

Figure 2.1
Figure 2.1. Figure 2.1: Our system X is partitioned into subsystems (V ,V⊥) at t = −1 and (Y ,Y⊥) at t = 0. We can also think of any part of our system as a “true insider” to its evolution, while thinking of any other part as extrinsic to the conscious part (assuming there is one) of X. Our task is to identify which of these perspectives matters most, and in what way, to the system X. To formalise this perspective, we define on… view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Applying the cause function ζV ,Y : V −1 ⊥ = v⊥ becomes a background condition, Y 0 ⊥ is discarded, and the remaining subsystems are partitioned. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Applying the effect function ζY ,V : Y 0 ⊥ = y⊥ becomes a background condition, V 1 ⊥ is discarded, and the remaining subsystems are partitioned. Definition 5 (Unconstrained Effect Repertoire). The unconstrained effect repertoire of a subsystem V with respect to subsystem Y (for a system X which has been observed as X0 = x 0 ), is the probability distribution of V 1 given that Y 0 is uniformly distribute… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Here we have a transition network which shows non-independent relationships between the inputs [PITH_FULL_IMAGE:figures/full_fig_p036_4_1.png] view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Input nodes Y t ⊥ = (Xt 1 , Xt 4 ) have been fixed to background conditions y t ⊥, and output nodes Y t+1 ⊥ = (X t+1 1 , Xt+1 4 ) have been marginalized over, to select the subsidiary network GY over Y = (X2, X3). On the left we have the unconstrained case in which entropy across Y t ⊔ Y t+1 has been maximized. On the right hand side we have maximized entropy subject to the conditions Y t+1 = y t+1, retr… view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: In this case, Y = (X2, X3) and our partition is P = {Xt 2 ⊔ X t+1 2 , Xt 3 ⊔ X t+1 3 }. The subsidiary network G y t Z|Y conditions on Xt 3 = x t 3 and applies a MaxCal path ensemble over the network. In this case, Xt 2 should have a uniform distribution over its state space Ω2 while the value x t+1 2 of X t+1 2 should be fixed by the background conditions. For our network G y t Z⊥|Y we condition on Xt 2… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: We understand our generative model as a graph with a series of biases and weights which combine to [PITH_FULL_IMAGE:figures/full_fig_p055_5_1.png] view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Here, we have a graph G over the state space Ω = {1, 2, 3, 4, 5}. The adjacency matrix A represents connections between nodes. The degree d(x) represents the number of edges a node x belongs to, e.g. d(5) = 4. PGRW =   0 0 0 0 1 0 0 1 3 1 3 1 3 0 1 3 0 1 3 1 3 0 1 3 1 3 0 1 3 1 5 1 5 1 5 1 5 1 5   , πGRW = 1 15   1 3 3 3 5   Here, each x will have a conditional entropy value l… view at source ↗
Figure 6
Figure 6. Figure 6: figure 6.1 to illustrate, we may express the transition probabilities and stationary distribution of [PITH_FULL_IMAGE:figures/full_fig_p065_6.png] view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Note that the colors are scaled independently across heatmaps. [PITH_FULL_IMAGE:figures/full_fig_p068_6_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: displays heatmaps for [PITH_FULL_IMAGE:figures/full_fig_p068_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the skew and mean values of [PITH_FULL_IMAGE:figures/full_fig_p069_6.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Pipes and mazes constrain maximal path entropy by limiting movement in physical space. Water will [PITH_FULL_IMAGE:figures/full_fig_p074_7_1.png] view at source ↗
Figure 7
Figure 7. Figure 7: figure 7.1, is water flowing through a pipe. Left to its own devices, water would spread chaotically in all directions. [PITH_FULL_IMAGE:figures/full_fig_p074_7.png] view at source ↗
read the original 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 $\psi$ 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).

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proposes defining information 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, re-deriving IIT's phenomenological calculus from constrained entropy-maximization (CMEP). The framework bridges to active inference (dual to CMEP under Langevin dynamics) and shows ψ equivalent to prediction error under CLT for Markov chains and LDT for Ising models, with potential relevance to the hill-shaped trajectory of Φ in neuronal cultures.

Significance. If the claimed mappings hold without auxiliary constraints, the work supplies a physically grounded unification of IIT and FEP, grounding consciousness measures in thermodynamic path ensembles and offering a route to extend IIT beyond current regimes. The special-case equivalences to prediction error and the link to FDT violations are notable strengths if supported by explicit derivations.

major comments (2)
  1. [Abstract] Abstract: the central claim that IIT 3.0 repertoires 'emerge directly' from MaxCal variational principles is asserted without visible supporting equations, explicit mapping, or verification steps; this is load-bearing for the equivalence to prediction error and the bridge to FEP.
  2. [Abstract] Abstract: the definition of ψ is introduced as a proposal and then used to recover IIT quantities, but the equivalence to prediction error under CLT/LDT appears to depend on the specific form of the constraints on the path ensemble; without showing the general case is free of unstated auxiliary assumptions, the claimed generality risks circularity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and have revised the manuscript to improve the clarity and explicitness of the abstract and derivations where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that IIT 3.0 repertoires 'emerge directly' from MaxCal variational principles is asserted without visible supporting equations, explicit mapping, or verification steps; this is load-bearing for the equivalence to prediction error and the bridge to FEP.

    Authors: We agree the abstract is concise and does not display the full equations. The manuscript derives the repertoires explicitly in Sections 3–4 by applying the MaxCal variational principle to the constrained path measure and showing that the resulting marginals recover the IIT cause-effect repertoires. To address the concern, we have revised the abstract to include a one-sentence outline of the variational step and added a forward reference to the relevant sections and equations. revision: partial

  2. Referee: [Abstract] Abstract: the definition of ψ is introduced as a proposal and then used to recover IIT quantities, but the equivalence to prediction error under CLT/LDT appears to depend on the specific form of the constraints on the path ensemble; without showing the general case is free of unstated auxiliary assumptions, the claimed generality risks circularity.

    Authors: The definition of ψ is the general deviation from the MaxCal ensemble under the observed constraints; the CLT and LDT equivalences follow from the standard statements of those theorems applied to the fluctuation statistics of the path measure, without further auxiliary constraints. We have added a clarifying paragraph in the revised discussion that states the assumptions explicitly and sketches the derivation steps to remove any appearance of circularity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from an explicit definitional proposal.

full rationale

The paper explicitly proposes a new definition of information as the deviation ψ from a constrained maximum-caliber path ensemble and then derives the IIT 3.0 cause/effect repertoires as consequences of the MaxCal variational principles applied to that definition. The claimed equivalence to prediction error is restricted to specific limiting cases (CLT for Markov chains and LDT for Ising models) rather than asserted as a general identity. No equations are presented in which an IIT quantity is shown to equal a fitted parameter or a self-referential constraint by construction, and no load-bearing self-citation chain is invoked to justify the central mapping. The argument is therefore self-contained as a theoretical re-expression rather than a tautological reduction of outputs to inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 1 invented entities

The central claim rests on a newly proposed definition of information together with standard mathematical tools; no numerical free parameters are explicitly fitted in the abstract, but the finite time horizon and choice of constraints function as definitional choices.

free parameters (2)
  • finite time horizon
    Chosen as part of the path-ensemble definition; affects the deviation measure ψ.
  • constraints on the path ensemble
    Specific constraints are required for the maximum-caliber construction but not enumerated in the abstract.
axioms (3)
  • standard math Variational principles of maximum caliber (CMEP)
    Invoked to derive cause/effect repertoires directly from constrained entropy maximization.
  • standard math Central Limit Theorem for Markov chains
    Applied to establish equivalence between ψ and prediction error.
  • standard math Large deviations theory for Ising models
    Used to show equivalence of ψ to prediction error in the Ising case.
invented entities (1)
  • information ψ no independent evidence
    purpose: Deviation of realized dynamics from constrained maximum-caliber path ensemble, serving as the bridge quantity between IIT and FEP
    Newly defined in the paper; no independent falsifiable handle outside the proposed framework is stated.

pith-pipeline@v0.9.0 · 5586 in / 1572 out tokens · 32946 ms · 2026-05-14T21:07:21.965052+00:00 · methodology

discussion (0)

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Reference graph

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