# Expected free energy as an information constraint on the Bethe Lagrangian

arXiv: https://arxiv.org/abs/2608.17167 · HTML: https://mlsanity.com/paper/2608.17167/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 24 references that are in the map.
- First posted on arXiv 17 Aug 2026.

## Builds on (most cited first, 10 of 10)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Exploration in Deep Reinforcement Learning: A Survey](https://mlsanity.com/paper/2205.00824/) (2022)
- [Active inference on discrete state-spaces: a synthesis](https://mlsanity.com/paper/2001.07203/) (2020)
- [From pixels to planning: scale-free active inference](https://mlsanity.com/paper/2407.20292/) (2024)
- [Expected Free Energy-based Planning as Variational Inference](https://mlsanity.com/paper/2504.14898/) (2025)
- [Realising Synthetic Active Inference Agents, Part II: Variational Message Updates](https://mlsanity.com/paper/2306.02733/) (2023)
- [Active Inference and Epistemic Value in Graphical Models](https://mlsanity.com/paper/2109.00541/) (2021)
- [Chance-Constrained Active Inference](https://mlsanity.com/paper/2102.08792/) (2021)
- [A Message Passing Realization of Expected Free Energy Minimization](https://mlsanity.com/paper/2508.02197/) (2025)
- [Message passing-based inference in an autoregressive active inference agent](https://mlsanity.com/paper/2509.25482/) (2025)

## Abstract (arXiv)

Active inference selects actions by minimising an expected free energy functional over predicted futures. However, adding an expectation over yet-unobserved outcomes means the free energy functional no longer has a Kullback-Leibler structure, which hinders message passing treatments of inference procedures. We propose an alternative formulation based on a Bethe free energy functional, fully supporting inference by message passing. The epistemic drive is maintained by imposing an information constraint, next to normalisation, marginalisation and form constraints, insisting that the mutual information between future observations, states and parameters given actions must be at least as large as the entropy of the goal prior. For a specific value of the corresponding Karush-Kuhn-Tucker multiplier, the stationary point of this constrained Bethe Lagrangian recovers the expected free energy solution. We show that, as the information demand is varied, the solved multiplier moves through its inactive, interior, and saturated regimes. In the inactive regime the agent's epistemic drive switches off entirely, while in the saturated regime it is maximal. We compare the performance of the constrained Bethe agent on three tasks against EFE and Q-MDP.