  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.
