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PreprintCited by 1Embodied and Extended CognitionGaussian Processes and Bayesian InferenceBayesian Modeling and Causal Inference

The generalized Hierarchical Gaussian Filter

Open access
Weber, Lilian Aline, Waade, Peter Thestrup, Legrand, Nicolas, Møller, Anna Hedvig, Stephan, Klaas Enno, Mathys, Christoph·2026·eLife
DOI 10.7554/elife.110174.1
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Abstract

Hierarchical Bayesian models of perception and learning feature prominently in contemporary cognitive neuroscience where, for example, they inform computational concepts of mental disorders. This includes predictive coding and hierarchical Gaussian filtering (HGF), which differ in the nature of hierarchical representations. In this work, we present a new class of artificial neural networks that unifies computational principles of PC and HGFs. We extend the space of generative models underlying HGF to include a form of nonlinear hierarchical coupling between state values akin to predictive coding and artificial neural networks in general. We derive the update equations corresponding to this generalization of HGF and conceptualize them as connecting a network of (belief) nodes where parent nodes either predict the state of child nodes or their rate of change. This enables us to (1) create modular architectures with generic computational steps in each node of the network, and (2) disclose the hierarchical message passing implied by generalized HGF models and to compare this to comparable schemes under predictive coding. The practical advances of this work are twofold: on the one hand, our extension allows for a modular construction of ANNs of arbitrarily complex hierarchical structure under the general principles of HGF. On the other hand, by providing a highly flexible implementation of hierarchical Bayesian models available as open source software, it enables new types of empi…

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  • Predictive coding networks can be reformulated as deep hierarchical Gaussian filters that replace iterative state inference with closed-form variational updates.

    “We close this gap by expressing predictive coding networks as generalised hierarchical Gaussian filters (gHGF)”

    From Closed-form predictive coding via hierarchical Gaussian filters (2026) · cites this paper as Weber et al., 2026

  • The gHGF tracks the posterior up to second order, updating both the mean and precision of every belief at each step, whereas standard predictive coding schemes optimize only the first moment and treat precisions as fixed or slowly learned hyperparameters.

    “The gHGF tracks the posterior up to second order, updating both the mean µ and the precision π of every belief at each step, whereas standard PC schemes optimize only the first moment and treat precisions as fixed or slowly learned hyperparameters.”

    From Closed-form predictive coding via hierarchical Gaussian filters (2026) · cites this paper as Weber et al., 2026

  • The gHGF natively supports volatility coupling, in which a parent modulates the rate of change rather than the mean of its child.

    “The gHGF natively supports volatility (and noise) coupling, in which a parent modulates the rate of change rather than the mean of its child via”

    From Closed-form predictive coding via hierarchical Gaussian filters (2026) · cites this paper as Weber et al., 2026

  • Modulation of precision by prediction error itself requires volatility coupling.

    “Modulation of precision by prediction error itself, the mechanism by which surprising observations should inflate uncertainty, requires volatility coupling [Weber et al., 2026].”

    From Closed-form predictive coding via hierarchical Gaussian filters (2026) · cites this paper as Weber et al., 2026

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Authors
Weber, L. A., Waade, P. T., Legrand, N., Møller, A. H., Stephan, K. E., & Mathys, C.
Affiliations
  • Osnabrück University (DE)
  • Aarhus University (DK)
  • University of Zurich (CH)
  • Scuola Internazionale Superiore di Studi Avanzati (IT)
ORCID
Venue
eLife
Publisher
eLife Sciences Publications, Ltd
Published
2026-03-02
ISSN
2050-084X (ISSN-L)
DOI
10.7554/elife.110174.1
OpenAlex
W7133245454
Cited by
0
References
18
Type
posted content
Language
en
Open access
Licence
Locations
eLife (acceptedVersion, cc-by)
Funders
  • René and Susanne Braginsky Foundation
  • Aarhus Universitets Forskningsfond (AUFF-E-2019-7-10)
  • Carlsbergfondet (CF21-0439)
Topics
  • Life Sciences › Neuroscience › Cognitive Neuroscience › Embodied and Extended Cognition
  • Physical Sciences › Computer Science › Artificial Intelligence › Gaussian Processes and Bayesian Inference
  • Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
Keywords
hierarchical Gaussian filter, predictive coding, Bayesian hierarchical model, generative models, artificial neural networks, modular network architecture, belief propagation, computational psychiatry

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