In our previous paper, we proposed a non-Gaussian Bayesian filter using power moments of the system state. A density surrogate parameterized as an analytic function is proposed to approximate the true system state, of which the distribution is only assumed Lebesgue integrable. To our knowledge, it is the first Bayesian filter where there is no prior constraints on the true density of the state and the state estimate has a continuous form of function. In this very preliminary version of paper, we propose a new type of statistics, which is called the generalized logarithmic moments. They are used to parameterize the state distribution together with the power moments. The map from the parameters of the proposed density surrogate to the power moments is proved to be a diffeomorphism, which allows to use gradient methods to treat the optimization problem determining the parameters. The simulation results reveal the advantage of using both moments for estimating mixtures of complicated types of functions.

翻译：在前期工作中，我们提出了一种基于系统状态幂矩的非高斯贝叶斯滤波器。通过构造参数化的解析函数密度代理来逼近真实系统状态，其中真实分布仅假定为勒贝格可积函数。据我们所知，这是首个无需对状态真实密度施加先验约束、且状态估计具有连续函数形式的贝叶斯滤波器。在本文初稿中，我们提出了一种新型统计量——广义对数矩，该统计量与幂矩共同用于状态分布的参数化。我们证明了从所提出的密度代理参数到幂矩的映射是微分同胚，这使得梯度方法可用于求解参数确定的优化问题。仿真结果表明，联合使用两种矩估计复杂函数类型的混合分布具有显著优势。