  This paper considers the problem of robust iterative Bayesian smoothing in
nonlinear state-space models with additive noise using Gaussian approximations.
Iterative methods are known to improve smoothed estimates but are not
guaranteed to converge, motivating the development of more robust versions of
the algorithms. The aim of this article is to present Levenberg-Marquardt (LM)
and line-search extensions of the classical iterated extended Kalman smoother
(IEKS) as well as the iterated posterior linearisation smoother (IPLS). The
IEKS has previously been shown to be equivalent to the Gauss-Newton (GN)
method. We derive a similar GN interpretation for the IPLS. Furthermore, we
show that an LM extension for both iterative methods can be achieved with a
simple modification of the smoothing iterations, enabling algorithms with
efficient implementations. Our numerical experiments show the importance of
robust methods, in particular for the IEKS-based smoothers. The computationally
expensive IPLS-based smoothers are naturally robust but can still benefit from
further regularisation.
