# Predictive Coding with Bayesian Priors via Proximal Gradients

arXiv: https://arxiv.org/abs/2606.08374 · HTML: https://mlsanity.com/paper/2606.08374/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 12 of its 35 references that are in the map.
- First posted on arXiv 6 Jun 2026.

## Builds on (most cited first, 12 of 12)

- [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)
- [The free energy principle for action and perception: A mathematical review](https://mlsanity.com/paper/1705.09156/) (2017)
- [Training Neural Networks Without Gradients: A Scalable ADMM Approach](https://mlsanity.com/paper/1605.02026/) (2016)
- [Deep Neural Network Structures Solving Variational Inequalities](https://mlsanity.com/paper/1808.07526/) (2018)
- [Predictive Coding, Variational Autoencoders, and Biological Connections](https://mlsanity.com/paper/2011.07464/) (2020)
- [Time-Varying Convex Optimization: A Contraction and Equilibrium Tracking Approach](https://mlsanity.com/paper/2305.15595/) (2023)
- [Positive Competitive Networks for Sparse Reconstruction](https://mlsanity.com/paper/2311.03821/) (2023)
- [Proximal Gradient Dynamics: Monotonicity, Exponential Convergence, and Applications](https://mlsanity.com/paper/2409.10664/) (2024)
- [Euclidean Contractivity of Neural Networks with Symmetric Weights](https://mlsanity.com/paper/2302.13452/) (2023)
- [An Energy-Based Mechanism for Compositional Behavior](https://mlsanity.com/paper/2512.04745/) (2025)
- [On Weakly Contracting Dynamics for Convex Optimization](https://mlsanity.com/paper/2403.07572/) (2024)

## Abstract (arXiv)

We recast predictive coding as continuous-time proximal gradient descent applied to a regularized maximum-a-posteriori (MAP) objective. We study first a single-level problem and then a multi-level hierarchy. For the single-level problem, we show that proximal gradient descent is precisely a leaky firing-rate network: the membrane leak, the effective recurrent matrix, the local synaptic drive, and the static nonlinearity all follow from one optimization principle, and the resulting circuit is the one proposed by Rao and Ballard. The prior selects the nonlinearity through its proximal operator, and the likelihood precision sets the gain on the observation. For the hierarchy, we show that a classical variable-splitting relaxation of the deep MAP problem yields hierarchical predictive coding as the interconnection of local and distributed solvers. In probabilistic modeling terms, this relaxation replaces the directed generative chain by an undirected Markov random field whose node potentials are the level-wise priors. Each level then applies its own activation function, namely the proximal operator of its prior.