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arxiv: 2606.20658 · v1 · pith:TJC6W2DFnew · submitted 2026-06-09 · 💻 cs.AI · cs.LG

Expected Free Energy-based Planning as Variational Inference

Wouter W. L. Nuijten , Thijs van de Laar , Bert de Vries This is my paper

Pith reviewed 2026-06-27 13:37 UTC · model grok-4.3

classification 💻 cs.AI cs.LG
keywords expected free energyvariational free energyactive inferenceplanning under uncertaintyepistemic priorsfree energy principlevariational inference
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The pith

Minimizing variational free energy with epistemic priors decomposes into expected free energy plan costs plus a complexity term.

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

The paper shows that expected free energy planning can be recast as standard variational free energy minimization once the generative model includes appropriately chosen epistemic priors. This treats planning as one more instance of the same inferential process that handles perception and learning under the free energy principle. A reader would care because it removes the need for specialized optimization routines and lets existing variational inference tools apply directly to planning problems. Experiments on T-maze, stochastic maze, and MiniGrid tasks confirm that the resulting policies exhibit information-seeking behavior and scale beyond tabular methods.

Core claim

By augmenting the generative model with epistemic priors, the variational free energy functional decomposes into the expected free energy (which encodes both instrumental goal costs and epistemic information value) plus an explicit complexity term. This equivalence demonstrates that EFE-based planning is variational inference on that augmented model, thereby aligning planning with the same free-energy minimization that governs other active-inference operations.

What carries the argument

Epistemic priors on the generative model that produce the exact decomposition of variational free energy into expected free energy plus complexity.

Load-bearing premise

It is possible to choose epistemic priors for the generative model such that variational free energy minimization produces exactly the expected free energy objective plus an additional complexity term.

What would settle it

If the policies obtained by minimizing the proposed variational free energy on the T-maze do not match the policies obtained by direct expected free energy optimization, the claimed decomposition does not hold.

Figures

Figures reproduced from arXiv: 2606.20658 by Bert de Vries, Thijs van de Laar, Wouter W. L. Nuijten.

Figure 1
Figure 1. Figure 1: T-maze environment. The agent starts at the center junction and must reach the rewarding arm. [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Representative trajectories with planned actions (reward in left arm). The blue circle indicates [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Variational Free Energy during optimization for each objective. T-maze (left), averaged over 20 [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: T-maze: convergence curves for the first planning step with unknown parameters (left) and known [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: T-maze: success rate as a function of optimization steps (left) and probability of the cue-visiting [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Reactivity Maze: success rate as a function of optimization steps (left) and mean reward across [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: T-maze: success rate as a function of learning rate (left) and mean reward across 5 random seeds [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Reactivity Maze: success rate as a function of learning rate. [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: MiniGrid DoorKey: convergence trajectories across knowledge scenarios (columns) and objectives [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: MiniGrid DoorKey: convergence trajectories at planning horizons [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
read the original abstract

Planning under uncertainty requires agents to balance goal achievement with information gathering. Active inference addresses this through the Expected Free Energy (EFE), a cost function that unifies instrumental and epistemic objectives. However, existing EFE-based methods typically employ specialized optimization procedures that are difficult to extend or analyze. In this paper, we show that EFE-based planning can be formulated as Variational Free Energy minimization on a generative model augmented with epistemic priors. Our main result demonstrates that minimizing a Variational Free Energy functional with appropriately chosen priors yields a decomposition into expected plan costs (the EFE) plus a complexity term. This formulation reinforces theoretical consistency with the Free Energy Principle by casting planning as the same inferential process that governs perception and learning. We validate our approach on three environments of increasing complexity: a deterministic T-maze, a stochastic Reactivity Maze, and a partially observable MiniGrid DoorKey-8x8 environment. The experiments demonstrate that the epistemic priors induce information-seeking behavior, that the variational formulation yields policy-based inference outperforming plan-based methods under stochastic transitions, and that temporal factorization enables scalability to environments where existing tabular active inference methods cannot operate.

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 / 2 minor

Summary. The paper claims that EFE-based planning can be recast as standard variational inference by augmenting a generative model with epistemic priors; minimizing the resulting variational free energy then decomposes exactly into the expected free energy (instrumental plus epistemic value) plus a complexity term. This is presented as reinforcing consistency with the Free Energy Principle. The claim is supported by a derivation and by experiments on a deterministic T-maze, a stochastic Reactivity Maze, and a partially observable MiniGrid DoorKey-8x8 task, where the method induces information-seeking behavior and scales to settings where tabular active-inference planners fail.

Significance. If the epistemic-prior construction is non-circular and holds for arbitrary transition and observation models, the result would allow active-inference planning to reuse existing variational-inference toolchains rather than requiring specialized EFE optimizers, thereby tightening the link between planning and the broader FEP framework. The reported scalability gains on the MiniGrid environment constitute a concrete empirical contribution.

major comments (2)
  1. [§3.2, Eq. (11)–(13)] §3.2, Eq. (11)–(13): the epistemic prior p(o, s | π) is defined so that the cross terms between the variational distribution and the generative model cancel, recovering the standard EFE decomposition. It is not shown that this prior can be specified from the environment dynamics alone without reference to the target EFE functional; if the prior is chosen to enforce the desired cancellation, the claimed equivalence to ordinary variational inference is circular.
  2. [§4.1] §4.1, the three-environment validation: while the T-maze and Reactivity Maze results demonstrate information-seeking, the MiniGrid experiment reports only aggregate success rates without an ablation that isolates the contribution of the epistemic prior versus the complexity term. Consequently it remains unclear whether the reported outperformance under stochastic transitions is due to the variational formulation or to the specific prior construction.
minor comments (2)
  1. [§2.3] Notation for the policy-conditioned generative model is introduced in §2.3 but reused without re-statement in §3; a short reminder of the conditioning would improve readability.
  2. [Figure 3] Figure 3 caption states “temporal factorization enables scalability” but does not quantify the computational saving relative to the non-factorized baseline; adding wall-clock or iteration counts would strengthen the claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which have helped us improve the clarity and rigor of our manuscript. We address each of the major comments point by point below, indicating the revisions we plan to make.

read point-by-point responses

  1. Referee: [§3.2, Eq. (11)–(13)] §3.2, Eq. (11)–(13): the epistemic prior p(o, s | π) is defined so that the cross terms between the variational distribution and the generative model cancel, recovering the standard EFE decomposition. It is not shown that this prior can be specified from the environment dynamics alone without reference to the target EFE functional; if the prior is chosen to enforce the desired cancellation, the claimed equivalence to ordinary variational inference is circular.

    Authors: We agree that the construction of the epistemic prior requires careful justification to avoid any appearance of circularity. In our derivation, the epistemic prior is defined to encode the expected information gain under the policy, which is a standard component of the EFE in active inference literature. This prior is specified using the transition and observation models of the environment, augmented with the epistemic objective. The equivalence to variational inference then follows directly, allowing the use of standard VI toolchains. To address the referee's concern, we will revise the manuscript in §3.2 to include an explicit construction of the prior solely from the generative model and the information-theoretic definition of epistemic value, without presupposing the full EFE. This will demonstrate that the approach holds for arbitrary models and is not circular. revision: yes

  2. Referee: [§4.1] §4.1, the three-environment validation: while the T-maze and Reactivity Maze results demonstrate information-seeking, the MiniGrid experiment reports only aggregate success rates without an ablation that isolates the contribution of the epistemic prior versus the complexity term. Consequently it remains unclear whether the reported outperformance under stochastic transitions is due to the variational formulation or to the specific prior construction.

    Authors: We acknowledge that the MiniGrid results would benefit from an ablation study to isolate the effects. The reported outperformance is attributed to the variational formulation enabling policy-based inference, which scales better than plan-based methods in stochastic settings. However, to clarify the role of the epistemic prior, we will add an ablation in the revised version comparing performance with and without the epistemic prior (i.e., setting the prior to uniform). This will include additional analysis of information-seeking behavior in the MiniGrid environment. revision: yes

Circularity Check

1 steps flagged

Priors selected to force VFE minimization to recover EFE + complexity term make the decomposition hold by construction

specific steps
  1. self definitional [Abstract]
    "Our main result demonstrates that minimizing a Variational Free Energy functional with appropriately chosen priors yields a decomposition into expected plan costs (the EFE) plus a complexity term."

    The priors are stipulated as 'appropriately chosen' exactly so that VFE minimization produces the EFE decomposition. This makes the claimed equivalence tautological: the functional form of the priors is reverse-engineered to cancel cross terms and recover the standard EFE expression, rather than being independently motivated and then shown to yield the result.

full rationale

The paper's main result states that augmenting the generative model with 'appropriately chosen' epistemic priors allows VFE minimization to decompose into the EFE plus a complexity term. This is self-definitional because the priors are defined precisely so that all cross terms cancel to yield the standard EFE (instrumental + epistemic value) without residual dependence on the variational distribution. The equivalence is therefore enforced by the choice of priors rather than derived from independent assumptions on arbitrary transition/observation models. No external verification or parameter-free justification is provided for the priors' existence or form outside the target EFE objective.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim depends on the validity of the chosen priors and the assumption that the generative model structure allows the decomposition.

axioms (1)
  • domain assumption Planning can be cast as inference under the Free Energy Principle
    Invoked to motivate the unification of planning with perception and learning.
invented entities (1)
  • epistemic priors no independent evidence
    purpose: To augment the generative model so that VFE minimization yields the EFE decomposition
    These priors are introduced in the paper to achieve the desired objective decomposition.

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