Online Generalised Predictive Coding

arXiv stat.ML / 5/5/2026

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Key Points

The paper proposes an extension of generalised filtering to online settings by combining latent-state inference, parameter learning, and uncertainty estimation in a unified “triple estimation” framework.
It focuses on adapting Dynamic Expectation Maximisation (DEM) for online data assimilation by separating temporal scales, enabling slow updates of parameters/precisions alongside fast Bayesian updating of hidden states.
The authors derive the variational principles and procedures that govern this online DEM (ODEM) workflow.
Numerical experiments show ODEM can track latent states generated by a nonlinear (possibly chaotic) generative process, even when the assimilation model’s functional form differs from the true dynamics.
The approach is positioned as a neuro-mimetic predictive-coding-inspired method for online inference, learning, and uncertainty quantification in dynamic environments.

Abstract

This paper introduces an extension of generalised filtering for online applications. Generalised filtering refers to data assimilation schemes that jointly infer latent states, learn unknown model parameters, and estimate uncertainty in an integrated framework -- e.g., estimate state and observation noise -- at the same time (i.e., triple estimation). This framework appears across disciplines under different names, including variational Kalman-Bucy filtering in engineering, generalised predictive coding in neuroscience, and Dynamic Expectation Maximisation (DEM) in time-series analysis. Here, we specialise DEM for ``online'' data assimilation, through a separation of temporal scales. We describe the variational principles and procedures that allow one to assimilate data in a way that allows for a slow updating of parameters and precisions, which contextualise fast Bayesian belief updating about the dynamic hidden states. Using numerical studies, we demonstrate the validity of online DEM (ODEM) using a non-linear -- and potentially chaotic -- generative model, to show that the ODEM scheme can track the latent states of the generative process, even when its functional form differs fundamentally from the dynamics of the generative model. Framed from a neuro-mimetic predictive coding perspective, ODEM offers a biologically inspired solution to online inference, learning, and uncertainty estimation in dynamic environments.