We consider the Bayesian optimal filtering problem: i.e. 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 the approximation capabilities of this framework. We provide approximation error bounds for filtering in general non-compact domains. We also consider strong time-uniform approximation error bounds that guarantee good long-time performance. We discuss and illustrate a number of practical concerns and implications of these results.

翻译：我们考虑贝叶斯最优滤波问题：即从观测序列中估计潜在时间序列信号的某些条件统计量。传统方法通常依赖于使用假定或估计的状态转移模型与观测模型。相反，我们构建了一个通用的循环神经网络框架，并尝试直接学习从观测输入到所需估计量统计量的递归映射。本文主要关注该框架的逼近能力。我们给出了在一般非紧致域上滤波的逼近误差界，同时考虑了保证良好长期性能的强时间一致逼近误差界。我们讨论并阐述了这些结果在实际中的若干关注点与启示。