  Appropriately designing the proposal kernel of particle filters is an issue
of significant importance, since a bad choice may lead to deterioration of the
particle sample and, consequently, waste of computational power. In this paper
we introduce a novel algorithm adaptively approximating the so-called optimal
proposal kernel by a mixture of integrated curved exponential distributions
with logistic weights. This family of distributions, referred to as mixtures of
experts, is broad enough to be used in the presence of multi-modality or
strongly skewed distributions. The mixtures are fitted, via online-EM methods,
to the optimal kernel through minimisation of the Kullback-Leibler divergence
between the auxiliary target and instrumental distributions of the particle
filter. At each iteration of the particle filter, the algorithm is required to
solve only a single optimisation problem for the whole particle sample,
yielding an algorithm with only linear complexity. In addition, we illustrate
in a simulation study how the method can be successfully applied to optimal
filtering in nonlinear state-space models.
