  Non-Gaussian Bayesian filtering is a core problem in stochastic filtering.
The difficulty of the problem lies in parameterizing the state estimates.
However the existing methods are not able to treat it well. We propose to use
power moments to obtain a parameterization. Unlike the existing parametric
estimation methods, our proposed algorithm does not require prior knowledge
about the state to be estimated, e.g. the number of modes and the feasible
classes of function. Moreover, the proposed algorithm is not required to store
massive parameters during filtering as the existing nonparametric Bayesian
filters, e.g. the particle filter. The parameters of the proposed
parametrization can also be determined by a convex optimization scheme with
moments constraints, to which the solution is proved to exist and be unique. A
necessary and sufficient condition for all the power moments of the density
estimate to exist and be finite is provided. The errors of power moments are
analyzed for the density estimate being either light-tailed or heavy-tailed.
Error upper bounds of the density estimate for the one-step prediction are
proposed. Simulation results on different types of density functions of the
state are given, including the heavy-tailed densities, to validate the proposed
algorithm.
