  In this paper, we aim to propose a consistent non-Gaussian Bayesian filter of
which the system state is a continuous function. The distributions of the true
system states, and those of the system and observation noises, are only assumed
Lebesgue integrable with no prior constraints on what function classes they
fall within. This type of filter has significant merits in both theory and
practice, which is able to ameliorate the curse of dimensionality for the
particle filter, a popular non-Gaussian Bayesian filter of which the system
state is parameterized by discrete particles and the corresponding weights. We
first propose a new type of statistics, called the generalized logarithmic
moments. Together with the power moments, they are used to form a density
surrogate, parameterized as an analytic function, to approximate the true
system state. The map from the parameters of the proposed density surrogate to
both the power moments and the generalized logarithmic moments is proved to be
a diffeomorphism, establishing the fact that there exists a unique density
surrogate which satisfies both moment conditions. This diffeomorphism also
allows us to use gradient methods to treat the convex optimization problem in
determining the parameters. Last but not least, simulation results reveal the
advantage of using both sets of moments for estimating mixtures of complicated
types of functions. A robot localization simulation is also given, as an
engineering application to validate the proposed filtering scheme.
