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arxiv: 2605.20293 · v1 · pith:ERMPGQ24new · submitted 2026-05-19 · 💻 cs.LG · cs.AI· cs.NE

Closed-form predictive coding via hierarchical Gaussian filters

Aleksandrs Baskakovs , Sylvain Estebe , Kenneth Enevoldsen , Kristoffer Nielbo , Chris Mathys , Nicolas Legrand This is my paper

Pith reviewed 2026-05-21 07:54 UTC · model grok-4.3

classification 💻 cs.LG cs.AIcs.NE
keywords predictive codinghierarchical Gaussian filtersprecision-weighted updatesvariational free energyonline learningdeep neural networksHebbian learningdynamic uncertainty
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The pith

Predictive coding networks can be rewritten exactly as deep hierarchical Gaussian filters to recover closed-form precision-weighted updates without iterations or global error signals.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Current predictive coding fixes the precision matrix to the identity, which slows learning and hurts performance in deeper networks by discarding the weighted prediction errors that variational inference demands. The paper recasts these networks as deep hierarchical Gaussian filters so that the free-energy objective directly supplies local, closed-form rules for updating activations, weights, and precisions together. Because the updates remain Hebbian-compatible and precision-weighted at every layer, inference finishes without iterative relaxation or automatic differentiation. On image classification the resulting networks approach backpropagation in wall-clock cost per epoch, converge faster, and show clear gains on online, data-efficient, and concept-drift settings.

Core claim

Identifying predictive coding networks with deep hierarchical Gaussian filters restores the full variational message passing: each layer computes precision-weighted prediction errors that serve as both learning signals and dynamic uncertainty estimates. Under a single free-energy objective the network simultaneously adjusts activations, weights, and precisions using only local rules, with no global error broadcast and no need for iterative inference or automatic differentiation at any layer.

What carries the argument

Deep hierarchical Gaussian filters (HGFs) that model the network layers as stacked Gaussians whose precision-weighted prediction errors carry messages both upward and downward.

If this is right

  • Activations, weights, and precisions are learned simultaneously under one local free-energy objective.
  • Inference resolves in closed form at each layer without iterations or automatic differentiation.
  • Dynamic uncertainty estimates emerge at every layer as a direct byproduct of the updates.
  • On FashionMNIST the networks approach backpropagation wall-clock cost while converging in fewer epochs.
  • Clear advantages appear on online learning, data efficiency, and adaptation to concept drift.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The closed-form structure may allow predictive coding to scale to greater depths where fixed-precision versions degrade.
  • Explicit per-layer uncertainty could be exploited for robust decisions when input statistics change over time.
  • Hebbian-compatible rules open a direct path to neuromorphic hardware implementations that avoid global broadcasts.

Load-bearing premise

Predictive coding networks can be expressed exactly as deep hierarchical Gaussian filters so that the variational derivation supplies closed-form, precision-weighted updates without further approximations or iterative relaxation.

What would settle it

Build a deep network using the hierarchical Gaussian filter equations and train it on FashionMNIST; if the method still requires iterative relaxation inside layers, fails to produce dynamic precision estimates, or loses its reported speed and accuracy advantage over backpropagation, the central claim is falsified.

Figures

Figures reproduced from arXiv: 2605.20293 by Aleksandrs Baskakovs, Chris Mathys, Kenneth Enevoldsen, Kristoffer Nielbo, Nicolas Legrand, Sylvain Estebe.

Figure 1
Figure 1. Figure 1: Predictive coding networks as deep hierarchical Gaussian filters. We express predictive [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Test accuracy learning curves on FashionMNIST. Rows correspond to hidden depth [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Adaptation and efficiency benchmarks. (a) Online learning: test error vs. iteration (batch size 1, 200 samples per iteration). (b) Data efficiency: test error vs. training set size (64 epochs per run). (c) Concept drift: test error over 3,000 iterations; label mapping for classes 5–9 permuted every 64 iterations (alternating grey bands). (d) Drift recovery: mean test error aligned to drift events, averaged… view at source ↗
Figure 4
Figure 4. Figure 4: Activation precision as a gating mechanism for learning and inference in deep networks. [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Computational cost comparison across depth and width configurations. [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The manuscript reformulates predictive coding (PC) networks as deep hierarchical Gaussian filters (HGFs) to restore precision-weighted message passing, enabling simultaneous closed-form learning of activations, weights, and precisions under a single free-energy objective without iterations or automatic differentiation. It claims this yields dynamic uncertainty estimates and Hebbian-compatible local updates, with empirical results on FashionMNIST approaching backpropagation in wall-clock cost while outperforming on online, data-efficiency, and concept-drift tasks.

Significance. If the claimed exact equivalence holds, the work would advance biologically plausible alternatives to backpropagation by providing a unified variational framework with local, precision-aware updates and online adaptability. Strengths include the explicit restoration of precision terms from the variational derivation, the focus on non-stationary learning scenarios, and the provision of reproducible empirical comparisons on online and concept-drift benchmarks.

major comments (1)
  1. [§3] §3 (HGF-PC equivalence derivation): The central claim that PC networks can be exactly expressed as deep HGFs to yield closed-form, precision-weighted updates without iterations or additional approximations for general nonlinear activations (e.g., ReLU) is load-bearing but insufficiently justified. For non-quadratic free-energy terms induced by nonlinearities, the variational family typically requires moment-matching or linearization steps; if these are implicit, they contradict the 'no iterations' and 'exact variational derivation' assertions and must be explicitly stated with a concrete example for a single nonlinear layer.
minor comments (3)
  1. [Abstract, §5] Abstract and §5 (results): The statement that the method 'approaches backpropagation in epoch-level wall-clock cost' lacks a specific quantitative table or figure reference showing per-epoch timings and would benefit from error bars across multiple runs.
  2. Notation: The distinction between precision matrices at different hierarchical levels is introduced without a clear summary table or equation cross-reference, which could be clarified for readers unfamiliar with HGF message passing.
  3. Related work: The manuscript would benefit from explicit comparison to prior PC variants that incorporate learned precisions, even if they require iterations, to better isolate the contribution of the closed-form HGF mapping.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive feedback on our manuscript. We address the major comment regarding the derivation in section 3 below, and we will revise the manuscript accordingly to provide additional clarification.

read point-by-point responses

  1. Referee: [§3] §3 (HGF-PC equivalence derivation): The central claim that PC networks can be exactly expressed as deep HGFs to yield closed-form, precision-weighted updates without iterations or additional approximations for general nonlinear activations (e.g., ReLU) is load-bearing but insufficiently justified. For non-quadratic free-energy terms induced by nonlinearities, the variational family typically requires moment-matching or linearization steps; if these are implicit, they contradict the 'no iterations' and 'exact variational derivation' assertions and must be explicitly stated with a concrete example for a single nonlinear layer.

    Authors: We appreciate this observation and agree that the treatment of nonlinear activations in the equivalence derivation warrants explicit elaboration. In the HGF framework, nonlinear activations are handled via local linearization (first-order Taylor expansion around the current mean estimate), which is a standard technique that allows the variational updates to remain in closed form without requiring iterative optimization or automatic differentiation. This linearization is part of the exact variational derivation under the assumed Gaussian variational family and does not introduce additional iterations beyond the single forward pass. We will revise §3 to include a concrete step-by-step example for a single layer with ReLU activation, demonstrating how the precision-weighted messages and updates are computed in closed form. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper's core move is to recast predictive coding networks inside the hierarchical Gaussian filter formalism so that variational free-energy minimization directly supplies precision-weighted, closed-form updates for activations, weights, and precisions. Although one co-author has prior publications on HGFs, the present manuscript treats the HGF structure as an established modeling language rather than deriving the target PC rules from a self-citation alone. The abstract and claimed results (FashionMNIST wall-clock parity, online learning, concept-drift robustness) are presented as external empirical checks on the resulting algorithm. No equation is shown to be definitionally identical to a fitted parameter or to a prior result whose only support is an overlapping-author citation; the variational derivation is therefore not forced by construction and retains independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that PC networks are exactly equivalent to deep HGFs under variational free energy, plus standard Gaussian and mean-field assumptions; no new free parameters or invented entities are introduced in the abstract, but the equivalence itself functions as an untested modeling axiom.

axioms (2)
  • domain assumption Predictive coding networks can be exactly recast as deep hierarchical Gaussian filters whose variational free-energy objective yields closed-form precision-weighted updates.
    Invoked in the abstract to justify restoring precision-weighted message passing and eliminating iterations.
  • standard math Gaussian assumptions and mean-field factorization hold across layers for the message passing to remain local and closed-form.
    Required for the HGF equivalence and dynamic uncertainty estimates.

pith-pipeline@v0.9.0 · 5755 in / 1365 out tokens · 32918 ms · 2026-05-21T07:54:32.019063+00:00 · methodology

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Reference graph

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