# Whence the Expected Free Energy?

arXiv: https://arxiv.org/abs/2004.08128 · HTML: https://mlsanity.com/paper/2004.08128/

## Key facts

- Cited by 40 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Apr 2020.

## Built on by (most cited first, 30 of 40)

- [Active Inference in Robotics and Artificial Agents: Survey and Challenges](https://mlsanity.com/paper/2112.01871/) (2021)
- [pymdp: A Python library for active inference in discrete state spaces](https://mlsanity.com/paper/2201.03904/) (2022)
- [The Free Energy Principle for Perception and Action: A Deep Learning Perspective](https://mlsanity.com/paper/2207.06415/) (2022)
- [Branching Time Active Inference: the theory and its generality](https://mlsanity.com/paper/2111.11107/) (2021)
- [Exploration and preference satisfaction trade-off in reward-free learning](https://mlsanity.com/paper/2106.04316/) (2021)
- [R-AIF: Solving Sparse-Reward Robotic Tasks from Pixels with Active Inference and World Models](https://mlsanity.com/paper/2409.14216/) (2024)
- [Branching Time Active Inference: empirical study and complexity class analysis](https://mlsanity.com/paper/2111.11276/) (2021)
- [A Unified View on Solving Objective Mismatch in Model-Based Reinforcement Learning](https://mlsanity.com/paper/2310.06253/) (2023)
- [Prior Preference Learning from Experts:Designing a Reward with Active Inference](https://mlsanity.com/paper/2101.08937/) (2021)
- [Object-based active inference](https://mlsanity.com/paper/2209.01258/) (2022)
- [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)
- [Neuroscience-inspired perception-action in robotics: applying active inference for state estimation, control and self-perception](https://mlsanity.com/paper/2105.04261/) (2021)
- [Bayesian neural networks for predicting uncertainty in full-field material response](https://mlsanity.com/paper/2406.14838/) (2024)
- [On Predictive planning and counterfactual learning in active inference](https://mlsanity.com/paper/2403.12417/) (2024)
- [Factorised Active Inference for Strategic Multi-Agent Interactions](https://mlsanity.com/paper/2411.07362/) (2024)
- [Active Inference and Human--Computer Interaction](https://mlsanity.com/paper/2412.14741/) (2024)
- [Online reinforcement learning with sparse rewards through an active inference capsule](https://mlsanity.com/paper/2106.02390/) (2021)
- [Successor Representation Active Inference](https://mlsanity.com/paper/2207.09897/) (2022)
- [Learning Policies for Continuous Control via Transition Models](https://mlsanity.com/paper/2209.08033/) (2022)
- [Sophisticated Learning: A novel algorithm for active learning during model-based planning](https://mlsanity.com/paper/2308.08029/) (2023)
- [Free Energy Projective Simulation (FEPS): Active inference with interpretability](https://mlsanity.com/paper/2411.14991/) (2024)
- [Object-centric proto-symbolic behavioural reasoning from pixels](https://mlsanity.com/paper/2411.17438/) (2024)
- [What Type of Inference is Active Inference?](https://mlsanity.com/paper/2606.04935/) (2026)
- [Realising Active Inference in Variational Message Passing: the Outcome-blind Certainty Seeker](https://mlsanity.com/paper/2104.11798/) (2021)
- [Advancing Perception in Artificial Intelligence through Principles of Cognitive Science](https://mlsanity.com/paper/2310.08803/) (2023)
- [Prior preferences in active inference agents: soft, hard, and goal shaping](https://mlsanity.com/paper/2512.03293/) (2025)
- [Curvature-aware Expected Free Energy as an Acquisition Function for Bayesian Optimization](https://mlsanity.com/paper/2603.26339/) (2026)
- [Active Inference as a Convex Markov Decision Process](https://mlsanity.com/paper/2607.20152/) (2026)
- [A Concise Mathematical Description of Active Inference in Discrete Time](https://mlsanity.com/paper/2406.07726/) (2024)

## 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.