In estimation theory, the Kushner equation provides the evolution of the probability density of the state of a dynamical system given continuous-time observations. Building upon our recent work, we propose a new way to approximate the solution of the Kushner equation through tractable variational Gaussian approximations of two proximal losses associated with the propagation and Bayesian update of the probability density. The first is a proximal loss based on the Wasserstein metric and the second is a proximal loss based on the Fisher metric. The solution to this last proximal loss is given by implicit updates on the mean and covariance that we proposed earlier. These two variational updates can be fused and shown to satisfy a set of stochastic differential equations on the Gaussian's mean and covariance matrix. This Gaussian flow is consistent with the Kalman-Bucy and Riccati flows in the linear case and generalize them in the nonlinear one.

翻译：在估计理论中，Kushner方程描述了给定连续时间观测下动态系统状态概率密度的演化过程。基于我们近期的研究工作，本文提出了一种新方法，通过两种与概率密度传播及贝叶斯更新相关的近端损失的易处理变分高斯近似来逼近Kushner方程的解。第一种近端损失基于Wasserstein度量，第二种近端损失基于Fisher度量。第二种近端损失的解由我们先前提出的均值和协方差隐式更新给出。这两种变分更新可融合并证明满足关于高斯分布的均值与协方差矩阵的一组随机微分方程。在线性情形下，该高斯流与Kalman-Bucy流和Riccati流一致，并在非线性情形下对其进行了推广。