A tutorial on the free-energy framework for modelling perception and learning
Open accessAbstract
This paper provides an easy to follow tutorial on the free-energy framework for modelling perception developed by Friston, which extends the predictive coding model of Rao and Ballard. These models assume that the sensory cortex infers the most likely values of attributes or features of sensory stimuli from the noisy inputs encoding the stimuli. Remarkably, these models describe how this inference could be implemented in a network of very simple computational elements, suggesting that this inference could be performed by biological networks of neurons. Furthermore, learning about the parameters describing the features and their uncertainty is implemented in these models by simple rules of synaptic plasticity based on Hebbian learning. This tutorial introduces the free-energy framework using very simple examples, and provides step-by-step derivations of the model. It also discusses in more detail how the model could be implemented in biological neural circuits. In particular, it presents an extended version of the model in which the neurons only sum their inputs, and synaptic plasticity only depends on activity of pre-synaptic and post-synaptic neurons. • Bayesian inference about stimulus properties can be performed by networks of neurons. • Learning about statistics of stimuli can be achieved by Hebbian synaptic plasticity. • Structure of the model resembles the hierarchical organization of the neocortex.
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The variational free energy decomposes into a sum of layer-wise, precision-weighted squared prediction errors, and modulating precisions dynamically weights the influence of each error signal on belief updates and learning.
“The variational free energy decomposes into a sum of layer-wise, precisionweighted squared prediction errors, and modulating precisions amounts to dynamically weighting the influence of each error signal on belief updates and learning [Friston, 2005, Bogacz, 2017].”
From Closed-form predictive coding via hierarchical Gaussian filters (2026) · cites this paper as Bogacz, 2017
In “Scaling up the model of perception” (Eq. 37; page unavailable), the paper gives precision-weighted quadratic prediction-error terms, and “Introducing hierarchy” extends the model to multiple layers. The Discussion says dividing errors by noise variance changes their influence on other neurons; the paper also gives error-dependent learning rules. But Eq. 37 includes log-covariance terms, so free energy is not just a sum of squared errors when precisions change. (LLM, conf=0.86)
Experimental — accuracy not yet measuredExisting predictive coding models have not been fit to data, but their simulated neural dynamics display many features observed in real data, such as oscillatory tendencies, even in very small networks.
“While, to our knowledge, existing predictive coding models have not been fit to data, the simulated neural dynamics display many features that are observed in real data, such as oscillatory tendencies, even in very small networks (Bogacz, 2017).”
From The generalized Hierarchical Gaussian Filter (2026) · cites this paper as Bogacz, 2017
In “Simplest example of perception > A possible neural implementation” (page unavailable), Bogacz describes a small simulated network whose excitatory and inhibitory connections produce oscillatory tendencies. The paper does not establish that existing predictive coding models have never been fit to data or that their simulated dynamics reproduce many features observed in real neural data. (LLM, conf=0.86)
Experimental — accuracy not yet measuredMost predictive coding schemes only focus on optimizing the first moment of the posterior distribution for perceptual inference, although the variational approach allows approximation of the full posterior including its variance.
“In contrast, most predictive coding schemes only focus on the optimization of the first moment [mode or expectation] of the posterior distribution for perceptual inference [Friston, 2005; Bogacz, 2017], although the variational approach does allow approximation of the full posterior distribution including its variance.”
From The generalized Hierarchical Gaussian Filter (2026) · cites this paper as Bogacz, 2017
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
The citation lists haven’t been fetched yet
References 21 · Cited by 283
- Authors
- Bogacz, R.
- Venue
- Journal of Mathematical Psychology
- Publisher
- Elsevier BV
- Published
- 2017-02
- Volume
- 76
- Issue
- Pt B
- Pages
- 198-211
- ISSN
- 0022-2496 (ISSN-L)
- PMC
- PMC5341759
- OpenAlex
- W2237059274
- Cited by
- 283
- References
- 21
- Citations by year
- 2026: 16 · 2025: 24 · 2024: 22 · 2023: 35 · 2022: 35 · 2021: 46 · 2020: 50 · 2019: 26 · 2018: 17 · 2017: 9 · 2016: 3
- FWCI
- 5.81
- Type
- journal article
- Language
- en
- Open access
- hybrid
- doi.org
- Locations
- Journal of Mathematical Psychology (publishedVersion, cc-by)
- Europe PMC (PubMed Central) (publishedVersion, cc-by)
- PubMed Central (publishedVersion, cc-by)
- PubMed (publishedVersion)
- Funders
- Medical Research Council (MC UU 12024/5)
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Embodied and Extended Cognition
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Keywords
- free energy principle, predictive coding, Bayesian inference, perception, learning, Hebbian learning, synaptic plasticity, artificial neural networks, neocortex, hierarchy, sensory cortex
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
