Exact smoothing in hidden conditionally markov switching linear models

We consider the problem of the exact calculation of smoothing in hidden switching state-space systems. There is a hidden state-space chain X, the switching chain R, and the observed chain Y. In the classical, widely used conditionally Gaussian state-space linear model (CGSSLM) the exact calculation with complexity linear in time is not feasible and different approximations have to be made. Different alternative models, in which the exact calculations are feasible, have been proposed recently. The key difference between these models and the classical ones is that R is Markovian conditionally on Y in the recent models, while it is not in the classical ones. Moreover, these different models have been extended to models in which X is no longer necessarily Markovian conditionally on (R, Y). Here, we propose a further new extension of the latter models and we derive exact computation of posterior expectation as well as posterior variance-covariance matrix with complexity polynomial in time.