# Closed-form predictive coding via hierarchical Gaussian filters

arXiv: https://arxiv.org/abs/2605.20293 · HTML: https://mlsanity.com/paper/2605.20293/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 34 references that are in the map.
- First posted on arXiv 19 May 2026.

## Builds on (most cited first, 17 of 17)

- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Equinox: neural networks in JAX via callable PyTrees and filtered transformations](https://mlsanity.com/paper/2111.00254/) (2021)
- [Predictive Coding Approximates Backprop along Arbitrary Computation Graphs](https://mlsanity.com/paper/2006.04182/) (2020)
- [Predictive Coding: a Theoretical and Experimental Review](https://mlsanity.com/paper/2107.12979/) (2021)
- [Associative Memories via Predictive Coding](https://mlsanity.com/paper/2109.08063/) (2021)
- [Learning on Arbitrary Graph Topologies via Predictive Coding](https://mlsanity.com/paper/2201.13180/) (2022)
- [Benchmarking Predictive Coding Networks -- Made Simple](https://mlsanity.com/paper/2407.01163/) (2024)
- [Hybrid Predictive Coding: Inferring, Fast and Slow](https://mlsanity.com/paper/2204.02169/) (2022)
- [A Stable, Fast, and Fully Automatic Learning Algorithm for Predictive Coding Networks](https://mlsanity.com/paper/2212.00720/) (2022)
- [Predictive Coding beyond Gaussian Distributions](https://mlsanity.com/paper/2211.03481/) (2022)
- [μPC: Scaling Predictive Coding to 100+ Layer Networks](https://mlsanity.com/paper/2505.13124/) (2025)
- [Predictive Coding Networks and Inference Learning: Tutorial and Survey](https://mlsanity.com/paper/2407.04117/) (2024)
- [Towards the Training of Deeper Predictive Coding Neural Networks](https://mlsanity.com/paper/2506.23800/) (2025)
- [Bidirectional predictive coding](https://mlsanity.com/paper/2505.23415/) (2025)
- [The generalized Hierarchical Gaussian Filter](https://mlsanity.com/paper/2305.10937/) (2023)
- [pyhgf: A neural network library for predictive coding](https://mlsanity.com/paper/2410.09206/) (2024)
- [Robust volatility updates for Hierarchical Gaussian Filtering](https://mlsanity.com/paper/2605.00966/) (2026)

## Built on by (most cited first, 1 of 1)

- [Single State Update Predictive Coding training for Time Series Forecasting and Anomaly Detection](https://mlsanity.com/paper/2608.24697/) (2026)

## Abstract (arXiv)

Predictive coding (PC) offers a local and biologically grounded alternative to backpropagation in the training of artificial neural networks, yet to date, it remains slower, and performance degrades sharply as network depth increases. We trace 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 to be fast, local, and Bayesian. We close this gap by expressing predictive coding networks as deep hierarchical Gaussian filters (HGFs) and restore precision-weighted message passing, yielding dynamic uncertainty estimates and Hebbian-compatible update rules at every layer. The resulting networks can simultaneously learn activations, weights, and precisions under a single free-energy objective, with no global error signal, and resolve inference without requiring iterations or automatic differentiation. On FashionMNIST, our solution approaches backpropagation in epoch-level wall-clock cost while converging in fewer epochs, and outperforms it on online, data efficiency, and concept-drift tasks. We thus establish that closed-form variational inference with online precision learning provides a tractable foundation for deep predictive coding networks, retaining biological and interpretative advantages, without requiring iterative relaxation or global error signals.