  This article addresses online variational estimation in state-space models.
We focus on learning the smoothing distribution, i.e. the joint distribution of
the latent states given the observations, using a variational approach together
with Monte Carlo importance sampling. We propose an efficient algorithm for
computing the gradient of the evidence lower bound (ELBO) in the context of
streaming data, where observations arrive sequentially. Our contributions
include a computationally efficient online ELBO estimator, demonstrated
performance in offline and true online settings, and adaptability for computing
general expectations under joint smoothing distributions.
