Recurrent Neural Networks and Universal Approximation of Bayesian Filters
This work addresses the challenge of estimating latent time-series signals without relying on assumed models, offering theoretical guarantees for practical filtering applications.
The paper tackles the Bayesian optimal filtering problem by proposing a recurrent neural network framework to learn recursive mappings from observations to estimator statistics, and provides approximation error bounds for non-compact domains and strong time-uniform bounds for long-term performance.
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.