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.

翻译：本文介绍了一种将广义滤波扩展至在线应用的方法。广义滤波是指一种数据同化方案，能够在统一框架中同时推断潜在状态、学习未知模型参数并估计不确定性（例如估计状态噪声与观测噪声），即三重估计。该框架在不同学科中具有不同名称，包括工程学中的变分卡尔曼-布西滤波（Variational Kalman-Bucy filtering）、神经科学中的广义预测编码（Generalised Predictive Coding）以及时间序列分析中的动态期望最大化（Dynamic Expectation Maximisation, DEM）。本文通过时间尺度的分离，将DEM专门用于“在线”数据同化。我们描述了相应的变分原理与步骤，使数据同化能够以允许参数及精度缓慢更新的方式进行，从而为动态隐藏状态的快速贝叶斯信念更新提供上下文。通过数值研究，我们利用一个非线性（且可能混沌）的生成模型验证了在线DEM（ODEM）的有效性，结果表明ODEM方案能够追踪生成过程的潜在状态，即使其函数形式与生成模型的动力学存在根本差异。从神经拟态预测编码的视角来看，ODEM为动态环境中的在线推断、学习与不确定性估计提供了一种受生物学启发的解决方案。