Smoothers are algorithms for Bayesian time series re-analysis. Most operational smoothers rely either on affine Kalman-type transformations or on sequential importance sampling. These strategies occupy opposite ends of a spectrum that trades computational efficiency and scalability for statistical generality and consistency: non-Gaussianity renders affine Kalman updates inconsistent with the true Bayesian solution, while the ensemble size required for successful importance sampling can be prohibitive. This paper revisits the smoothing problem from the perspective of measure transport, which offers the prospect of consistent prior-to-posterior transformations for Bayesian inference. We leverage this capacity by proposing a general ensemble framework for transport-based smoothing. Within this framework, we derive a comprehensive set of smoothing recursions based on nonlinear transport maps and detail how they exploit the structure of state-space models in fully non-Gaussian settings. We also describe how many standard Kalman-type smoothing algorithms emerge as special cases of our framework. A companion paper (Ramgraber et al., 2023) explores the implementation of nonlinear ensemble transport smoothers in greater depth.

翻译：平滑器是用于贝叶斯时间序列再分析的算法。大多数业务化平滑器依赖于仿射卡尔曼型变换或序列重要性采样。这些策略处于一个频谱的两端，该频谱在计算效率和可扩展性与统计通用性和一致性之间进行权衡：非高斯性使仿射卡尔曼更新与真实贝叶斯解不一致，而成功进行重要性采样所需的集合规模可能高得令人望而却步。本文从测度传输的角度重新审视平滑问题，测度传输为贝叶斯推断中的先验到后验一致性变换提供了可能。我们利用这一能力，提出了一个基于传输的通用集合平滑框架。在该框架内，我们推导出一套基于非线性传输图的完整平滑递推公式，并详细说明了它们如何在完全非高斯设置中利用状态空间模型的结构。我们还描述了多种标准卡尔曼型平滑算法如何作为我们框架的特例出现。配套论文（Ramgraber等，2023）将更深入地探讨非线性集合传输平滑器的实现。