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.

翻译：非高斯贝叶斯滤波是随机滤波中的核心问题，其难点在于状态估计的参数化表征。现有方法难以有效处理该问题。本文提出利用幂矩实现参数化方案。与现有参数估计方法不同，该算法无需预知待估计状态的先验信息（如模态数量及可行函数类）。同时，相较于现有非参数贝叶斯滤波器（如粒子滤波器），所提算法在滤波过程中无需存储海量参数。通过引入矩约束的凸优化框架可确定参数化方案中的参数，且证明了该优化解的存在性与唯一性。给出了密度估计所有幂矩存在且有限的充要条件。针对轻尾与重尾两类密度估计情形，分析了幂矩误差特性，并提出了单步预测中密度估计的误差上界。通过包含重尾密度在内的多类状态密度函数仿真实验，验证了所提算法的有效性。