Variational free energy and the Laplace approximation
Abstract
This note derives the variational free energy under the Laplace approximation, with a focus on accounting for additional model complexity induced by increasing the number of model parameters. This is relevant when using the free energy as an approximation to the log-evidence in Bayesian model averaging and selection. By setting restricted maximum likelihood (ReML) in the larger context of variational learning and expectation maximisation (EM), we show how the ReML objective function can be adjusted to provide an approximation to the log-evidence for a particular model. This means ReML can be used for model selection, specifically to select or compare models with different covariance components. This is useful in the context of hierarchical models because it enables a principled selection of priors that, under simple hyperpriors, can be used for automatic model selection and relevance determination (ARD). Deriving the ReML objective function, from basic variational principles, discloses the simple relationships among Variational Bayes, EM and ReML. Furthermore, we show that EM is formally identical to a full variational treatment when the precisions are linear in the hyperparameters. Finally, we also consider, briefly, dynamic models and how these inform the regularisation of free energy ascent schemes, like EM and ReML.
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The ReML objective function and the variational energy are formally identical under this model.
“It is interesting to note that the ReML objective function and the variational energy are formally identical under this model [15],[18].”
From Hierarchical Models in the Brain (2008) · cites this paper as [15]
Static cases of the fixed-form scheme reduce to expectation maximisation (EM) and restricted maximum likelihood (ReML) for linear models.
“We have discussed static cases of this fixed-form scheme previously and how it reduces to expectation maximisation (EM; [17]) and restricted maximum likelihood (ReML; [18]) for linear models [15].”
From Hierarchical Models in the Brain (2008) · cites this paper as [15]
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Self-citation
- Authors
- Friston, K., Mattout, J., Trujillo-Barreto, N., Ashburner, J., & Penny, W.
- Affiliations
- UCL Queen Square Institute of Neurology (GB)
- Corresponding author
- Friston, Karl
- Venue
- NeuroImage
- Publisher
- Elsevier BV
- Published
- 2007-01
- Volume
- 34
- Issue
- 1
- Pages
- 220-234
- ISSN
- 1053-8119 (ISSN-L)
- OpenAlex
- W2113257799
- Cited by
- 880
- References
- 5
- Citations by year
- 2026: 41 · 2025: 46 · 2024: 41 · 2023: 57 · 2022: 64 · 2021: 90 · 2020: 73 · 2019: 50 · 2018: 46 · 2017: 41 · 2016: 45 · 2015: 51 · 2014: 49 · 2013: 26 · 2012: 34
- FWCI
- 27.35
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- NeuroImage (publishedVersion)
- PubMed (publishedVersion)
- UCL Discovery (University College London) (submittedVersion)
- Research Explorer (The University of Manchester) (submittedVersion)
- Funders
- Wellcome Trust
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Gaussian Processes and Bayesian Inference
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Bayesian Inference
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Methods and Mixture Models
- Keywords
- variational free energy, Laplace approximation, restricted maximum likelihood, variational inference, expectation-maximization algorithm, Bayesian model selection, Bayesian model averaging, hierarchical models, hyperprior, dynamic model, model complexity
- MeSH terms
- Bayes Theorem, Physics, Models, Statistical
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 ›
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