| name | closed-form-predictive-coding-hgf |
| description | Closed-form predictive coding via hierarchical Gaussian filters (HGF) methodology from arXiv:2605.20293. Restores precision-weighted prediction errors to predictive coding networks by expressing them as deep hierarchical Gaussian filters. Enables biologically plausible, local learning without backpropagation, with dynamic uncertainty estimates and Hebbian-compatible update rules. Activation: predictive coding, hierarchical Gaussian filter, free energy principle, precision-weighted prediction error, biologically plausible learning, HGF, predictive coding network. |
Closed-Form Predictive Coding via Hierarchical Gaussian Filters
Methodology from arXiv:2605.20293 (May 2026). Authors: Aleksandrs Baskakovs, Sylvain Estebe, Kenneth Enevoldsen, Kristoffer Nielbo, Chris Mathys, Nicolas Legrand.
Overview
Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in training artificial neural networks. However, current PC networks suffer from two key problems: they are slower than backpropagation, and performance degrades sharply as network depth increases.
This paper traces both problems to a single simplification: current PC networks fix the precision matrix to the identity, discarding precision-weighted prediction errors that the variational derivation requires. The authors close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing.
Key insight: The HGF framework provides closed-form variational updates for all parameters — activations, weights, and precisions — under a single free-energy objective, requiring no global error signal, no iterative relaxation, and no automatic differentiation.
Core Methodology
1. Precision-Weighted Predictive Coding
Traditional predictive coding networks use fixed identity precision matrices, which loses the precision-weighting that makes PC theoretically grounded. This paper restores precision-weighting by expressing PC networks as hierarchical Gaussian filters.
2. Hierarchical Gaussian Filter (HGF) Framework
