Ensemble transport smoothing. Part I: Unified framework

TL;DR

This paper introduces a unified ensemble transport framework for Bayesian smoothing in time series analysis, capable of handling non-Gaussian models more consistently than traditional methods.

Contribution

It proposes a general transport-based smoothing framework that unifies and extends existing Kalman and importance sampling smoothers for non-Gaussian settings.

Findings

Derives new smoothing recursions using nonlinear transport maps.

Shows how standard Kalman smoothers are special cases of the framework.

Provides a foundation for more consistent Bayesian inference in complex models.

Abstract

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.