# Active inference on discrete state-spaces: a synthesis

arXiv: https://arxiv.org/abs/2001.07203 · HTML: https://mlsanity.com/paper/2001.07203/

## Key facts

- Cited by 92 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2020.

## Built on by (most cited first, 30 of 92)

- [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)
- [Designing Ecosystems of Intelligence from First Principles](https://mlsanity.com/paper/2212.01354/) (2022)
- [Contrastive Active Inference](https://mlsanity.com/paper/2110.10083/) (2021)
- [Branching Time Active Inference: the theory and its generality](https://mlsanity.com/paper/2111.11107/) (2021)
- [On efficient computation in active inference](https://mlsanity.com/paper/2307.00504/) (2023)
- [An empirical evaluation of active inference in multi-armed bandits](https://mlsanity.com/paper/2101.08699/) (2021)
- [Exploration and preference satisfaction trade-off in reward-free learning](https://mlsanity.com/paper/2106.04316/) (2021)
- [Active Inference as a Model of Agency](https://mlsanity.com/paper/2401.12917/) (2024)
- [Understanding the Origin of Information-Seeking Exploration in Probabilistic Objectives for Control](https://mlsanity.com/paper/2103.06859/) (2021)
- [Expected Free Energy-based Planning as Variational Inference](https://mlsanity.com/paper/2504.14898/) (2025)
- [Applications of the Free Energy Principle to Machine Learning and Neuroscience](https://mlsanity.com/paper/2107.00140/) (2021)
- [Branching Time Active Inference: empirical study and complexity class analysis](https://mlsanity.com/paper/2111.11276/) (2021)
- [Hierarchical generative modelling for autonomous robots](https://mlsanity.com/paper/2308.07775/) (2023)
- [A Unified View on Solving Objective Mismatch in Model-Based Reinforcement Learning](https://mlsanity.com/paper/2310.06253/) (2023)
- [Supervised structure learning](https://mlsanity.com/paper/2311.10300/) (2023)
- [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)
- [Bayesian brains and the R'enyi divergence](https://mlsanity.com/paper/2107.05438/) (2021)
- [Active Inference for Stochastic Control](https://mlsanity.com/paper/2108.12245/) (2021)
- [Active Inference and Epistemic Value in Graphical Models](https://mlsanity.com/paper/2109.00541/) (2021)
- [Active inference, Bayesian optimal design, and expected utility](https://mlsanity.com/paper/2110.04074/) (2021)
- [Resolving uncertainty on the fly: Modeling adaptive driving behavior as active inference](https://mlsanity.com/paper/2311.06417/) (2023)
- [Neuroscience-inspired perception-action in robotics: applying active inference for state estimation, control and self-perception](https://mlsanity.com/paper/2105.04261/) (2021)
- [Geometric Methods for Sampling, Optimisation, Inference and Adaptive Agents](https://mlsanity.com/paper/2203.10592/) (2022)
- [Deconstructing deep active inference](https://mlsanity.com/paper/2303.01618/) (2023)
- [A Message Passing Realization of Expected Free Energy Minimization](https://mlsanity.com/paper/2508.02197/) (2025)
- [Realising Synthetic Active Inference Agents, Part I: Epistemic Objectives and Graphical Specification Language](https://mlsanity.com/paper/2306.08014/) (2023)
- [Integrating cognitive map learning and active inference for planning in ambiguous environments](https://mlsanity.com/paper/2308.08307/) (2023)

## Abstract (arXiv)

Active inference is a normative principle underwriting perception, action, planning, decision-making and learning in biological or artificial agents. From its inception, its associated process theory has grown to incorporate complex generative models, enabling simulation of a wide range of complex behaviours. Due to successive developments in active inference, it is often difficult to see how its underlying principle relates to process theories and practical implementation. In this paper, we try to bridge this gap by providing a complete mathematical synthesis of active inference on discrete state-space models. This technical summary provides an overview of the theory, derives neuronal dynamics from first principles and relates this dynamics to biological processes. Furthermore, this paper provides a fundamental building block needed to understand active inference for mixed generative models; allowing continuous sensations to inform discrete representations. This paper may be used as follows: to guide research towards outstanding challenges, a practical guide on how to implement active inference to simulate experimental behaviour, or a pointer towards various in-silico neurophysiological responses that may be used to make empirical predictions.