Recurrent Neural Networks and (Time-Uniform) Universal Approximation of Bayesian Filters

• 成果类型: Article
• 署名作者: Bishop, Adrian N.; Bonilla, Edwin V.; Del Moral, Pierre
• 署名单位: Commonwealth Scientific & Industrial Research Organisation (CSIRO); Commonwealth Scientific & Industrial Research Organisation (CSIRO); CSIRO Data61; Inria
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2026.3655312
• 发表日期: 2026
• 关键词: FOURIER-HERMITE EXPANSION; nonlinear filters; STABILITY; propagation; chaos

摘要

We consider approximations of the Bayesian optimal filtering problem, e.g., estimating some conditional statistics of a latent time-series signal from an observation sequence. Classical approaches often rely on the use of assumed or estimated transition and observation models. Instead, we formulate a generic recurrent neural network framework and seek to learn directly a recursive mapping from observational inputs to the desired estimator statistics. The main focus of this article is on the approximation capabilities of this framework. We provide approximation error bounds for filtering in general noncompact domains. Importantly, we also consider strong time-uniform approximation error bounds that guarantee good long-time performance, i.e., we show under certain conditions that filter approximation errors do not accumulate unbounded over time. We discuss and illustrate a number of practical concerns and implications of these results.