Exact Fast Computation of Optimal Filter in Gaussian Switching Linear Systems

We consider triplet Markov Gaussian linear systems (X, R, Y), where X is hidden continuous random sequence, R is hidden discrete Markov chain, Y is observed continuous random sequence, and (X, Y) is Gaussian conditionally on R. In the classical “Conditionally Gaussian Linear State-Space Model” (CGLSSM), optimal filter is not workable with a reasonable complexity. The aim of the paper is to propose a new model, quite close to the CGLSSM, belonging to the general and recently proposed family of models, called “Conditionally Markov Switching Hidden Linear Models” (CMSHLMs), in which the computation of optimal filter with complexity linear in the number of observations is feasible. The new model and related filtering are immediately applicable in all situations where the classical CGLSSM is used via approximated filtering.