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.

翻译：本文研究加性噪声非线性状态空间模型中基于高斯近似的鲁棒迭代贝叶斯平滑问题。迭代方法虽能改善平滑估计，但收敛性无法保证，这促使研究者开发更鲁棒的算法版本。本文旨在提出经典迭代扩展卡尔曼平滑器（IEKS）与迭代后验线性化平滑器（IPLS）的Levenberg-Marquardt（LM）扩展与线性搜索扩展。已有研究表明IEKS等价于高斯-牛顿（GN）方法。我们推导出IPLS的类似GN解释。进一步证明，通过对平滑迭代进行简单修改即可实现两种迭代方法的LM扩展，从而获得具备高效实现能力的算法。数值实验表明鲁棒方法的重要性，尤其对于基于IEKS的平滑器而言。计算成本高昂的基于IPLS的平滑器虽具有天然鲁棒性，但仍可从额外正则化中获益。