  Sequential Bayesian Filtering aims to estimate the current state distribution
of a Hidden Markov Model, given the past observations. The problem is
well-known to be intractable for most application domains, except in notable
cases such as the tabular setting or for linear dynamical systems with gaussian
noise. In this work, we propose a new class of filters based on Gaussian PSD
Models, which offer several advantages in terms of density approximation and
computational efficiency. We show that filtering can be efficiently performed
in closed form when transitions and observations are Gaussian PSD Models. When
the transition and observations are approximated by Gaussian PSD Models, we
show that our proposed estimator enjoys strong theoretical guarantees, with
estimation error that depends on the quality of the approximation and is
adaptive to the regularity of the transition probabilities. In particular, we
identify regimes in which our proposed filter attains a TV $\epsilon$-error
with memory and computational complexity of $O(\epsilon^{-1})$ and
$O(\epsilon^{-3/2})$ respectively, including the offline learning step, in
contrast to the $O(\epsilon^{-2})$ complexity of sampling methods such as
particle filtering.
