Multi-scale problems, where variables of interest evolve in different time-scales and live in different state-spaces, can be found in many fields of science. Here, we introduce a new recursive methodology for Bayesian inference that aims at estimating the static parameters and tracking the dynamic variables of these kind of systems. Although the proposed approach works in rather general multi-scale systems, for clarity we analyze the case of a heterogeneous multi-scale model with 3 time-scales (static parameters, slow dynamic state variables and fast dynamic state variables). The proposed scheme, based on nested filtering methodology of P\'erez-Vieites et al. (2018), combines three intertwined layers of filtering techniques that approximate recursively the joint posterior probability distribution of the parameters and both sets of dynamic state variables given a sequence of partial and noisy observations. We explore the use of sequential Monte Carlo schemes in the first and second layers while we use an unscented Kalman filter to obtain a Gaussian approximation of the posterior probability distribution of the fast variables in the third layer. Some numerical results are presented for a stochastic two-scale Lorenz 96 model with unknown parameters.

翻译：多尺度问题广泛存在于科学研究的诸多领域，其关注变量在不同时间尺度上演化且存在于不同的状态空间。本文提出一种新的贝叶斯推断递归方法，旨在估计此类系统的静态参数并跟踪其动态变量。尽管所提方法适用于相当广泛的多尺度系统，但为清晰起见，我们重点分析包含三个时间尺度（静态参数、慢速动态状态变量与快速动态状态变量）的异构多尺度模型。该方案基于Pérez-Vieites等人（2018）的嵌套滤波方法，通过三个相互交织的滤波层进行递归逼近，以近似获得给定部分噪声观测序列条件下参数与两组动态状态变量的联合后验概率分布。我们在第一层和第二层探索使用序贯蒙特卡罗方法，同时在第三层采用无迹卡尔曼滤波器以获得快速变量后验概率分布的高斯近似。针对参数未知的随机双尺度Lorenz 96模型，我们展示了若干数值计算结果。