mlsanityData to 7 Oct 2026

Whence the Expected Free Energy?

arXiv 2004.08128 ↗First posted 17 Apr 2020cs.AIcited by 40

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Abstract (arXiv)

The Expected Free Energy (EFE) is a central quantity in the theory of active inference. It is the quantity that all active inference agents are mandated to minimize through action, and its decomposition into extrinsic and intrinsic value terms is key to the balance of exploration and exploitation that active inference agents evince. Despite its importance, the mathematical origins of this quantity and its relation to the Variational Free Energy (VFE) remain unclear. In this paper, we investigate the origins of the EFE in detail and show that it is not simply "the free energy in the future". We present a functional that we argue is the natural extension of the VFE, but which actively discourages exploratory behaviour, thus demonstrating that exploration does not directly follow from free energy minimization into the future. We then develop a novel objective, the Free-Energy of the Expected Future (FEEF), which possesses both the epistemic component of the EFE as well as an intuitive mathematical grounding as the divergence between predicted and desired futures.

Key facts. Cited by 40 papers in the map (data through 7 Oct 2026). First posted on arXiv 17 Apr 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 9; 2022: 5; 2023: 4; 2024: 9; 2025: 2; 2026: 11.

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  1. 2021Active Inference in Robotics and Artificial Agents: Survey and Challengescited by 52
  2. 2022pymdp: A Python library for active inference in discrete state spacescited by 39
  3. 2022The Free Energy Principle for Perception and Action: A Deep Learning Perspectivecited by 24
  4. 2021Branching Time Active Inference: the theory and its generalitycited by 15
  5. 2021Exploration and preference satisfaction trade-off in reward-free learningcited by 12
  6. 2024R-AIF: Solving Sparse-Reward Robotic Tasks from Pixels with Active Inference and World Modelscited by 10
  7. 2021Branching Time Active Inference: empirical study and complexity class analysiscited by 9
  8. 2023A Unified View on Solving Objective Mismatch in Model-Based Reinforcement Learningcited by 9
  9. 2021Prior Preference Learning from Experts:Designing a Reward with Active Inferencecited by 8
  10. 2022Object-based active inferencecited by 7
  11. 2023Realising Synthetic Active Inference Agents, Part II: Variational Message Updatescited by 7
  12. 2021Active Inference and Epistemic Value in Graphical Modelscited by 6
28 more
  1. 2021Neuroscience-inspired perception-action in robotics: applying active inference for state estimation, control and self-perceptioncited by 5
  2. 2024Bayesian neural networks for predicting uncertainty in full-field material responsecited by 5
  3. 2024On Predictive planning and counterfactual learning in active inferencecited by 4
  4. 2024Factorised Active Inference for Strategic Multi-Agent Interactionscited by 4
  5. 2024Active Inference and Human--Computer Interactioncited by 4
  6. 2021Online reinforcement learning with sparse rewards through an active inference capsulecited by 3
  7. 2022Successor Representation Active Inferencecited by 3
  8. 2022Learning Policies for Continuous Control via Transition Modelscited by 3
  9. 2023Sophisticated Learning: A novel algorithm for active learning during model-based planningcited by 2
  10. 2024Object-centric proto-symbolic behavioural reasoning from pixelscited by 2
  11. 2026What Type of Inference is Active Inference?cited by 2
  12. 2021Realising Active Inference in Variational Message Passing: the Outcome-blind Certainty Seekercited by 1
  13. 2023Advancing Perception in Artificial Intelligence through Principles of Cognitive Sciencecited by 1
  14. 2024Free Energy Projective Simulation (FEPS): Active inference with interpretabilitycited by 1
  15. 2026Curvature-aware Expected Free Energy as an Acquisition Function for Bayesian Optimizationcited by 1
  16. 2026Active Inference as a Convex Markov Decision Processcited by 1
  17. 2024A Concise Mathematical Description of Active Inference in Discrete Timecited by 0
  18. 2024Planning to avoid ambiguous states through Gaussian approximations to non-linear sensors in active inference agentscited by 0
  19. 2025NAEL: Non-Anthropocentric Ethical Logiccited by 0
  20. 2025Prior preferences in active inference agents: soft, hard, and goal shapingcited by 0
  21. 2026Optimizing Neurorobot Policy under Limited Demonstration Data through Preference Regretcited by 0
  22. 2026Prediction and Empowerment: A Theory of Agency through Bridge Interfacescited by 0
  23. 2026Mind Dreamer: Untethering Imagination via Active Causal Intervention on Latent Manifoldscited by 0
  24. 2026Expected Free Energy as Belief-Dependent Utility for rho-POMDPscited by 0
  25. 2026Generalised Bellman recurrence and three dualities in sequential decision-makingcited by 0
  26. 2026The Dually Flat Geometry of Planning as Inferencecited by 0
  27. 2026Math for AI safety: an invitation for mathematicianscited by 0
  28. 2026BOBA: Dynamic Bayesian Optimization through Bayesian Active Inferencecited by 0