  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.
