AI 中文总结
研究模型误设下贝叶斯滤波，通过预测算子和更新算子构建框架，推导最优与误设滤波器差异的显式界，分析滤波误差传播因素，给出解析示例，为量化非线性滤波器鲁棒性提供理论基础。
AI 中文摘要
我们研究模型动力学误设下的贝叶斯滤波。最优滤波器由与真实动力学相关的预测算子 \(K = \{K_t\}_{t\geq1}\) 控制，近似滤波器使用扰动算子 \(\widehat{K} = \{\widehat{K}_t\}_{t\geq1}\)。两个滤波器共享由观测模型诱导的相同更新算子 \(U = \{U_t\}_{t\geq1}\)。我们在完全概率设置下工作，其中观测是随机变量而非固定实现。这一观点为最优滤波器的均值测度产生线性递归，并使滤波误差的传播取决于两个因素：预测算子的收缩和累积模型差异。在此框架内，我们推导了最优滤波器和误设滤波器之间差异的显式界。这些界将初始条件误差的影响与动力学失配 \(\widehat{K} - K\) 引入的扰动分开。在合适的收缩性假设下，累积误差稳定在由误设大小和预测算子的多布鲁申系数决定的稳态水平。我们还给出了可直接验证假设的解析示例。特别是，如果转移函数有界且转移噪声属于包括高斯、学生 \(t\) 和拉普拉斯定律在内的一类椭圆对称分布，则滤波误差如理论预测那样稳定。结果为量化随机观测和误设动力学下非线性滤波器的鲁棒性提供了理论基础。
英文摘要
We study Bayesian filtering under misspecified model dynamics. The optimal filter is governed by predictive operators $K=\{K_t\}_{t\ge 1}$ associated with the true dynamics, while the approximate filter uses perturbed operators $\widehat K=\{\widehat K_t\}_{t\ge 1}$. Both filters share the same update operators $U=\{U_t\}_{t\ge 1}$ induced by the observation model. We work in a fully probabilistic setting in which observations are random variables, rather than fixed realizations. This viewpoint yields a linear recursion for the mean measures of the optimal filter and makes the propagation of filtering errors depend on two ingredients: the contraction of the predictive operators, and the accumulated model discrepancy. Within this framework, we derive explicit bounds for the discrepancy between the optimal and misspecified filters. The bounds separate the effect of the initial-condition error from the perturbation introduced by the dynamical mismatch $\widehat K-K$. Under suitable contractivity assumptions, the cumulative error stabilizes at a steady-state level determined by the size of the misspecification and by the Dobrushin coefficients of the predictive operators. We also give analytical examples in which the assumptions can be verified directly. In particular, if the transition functions are bounded and the transition noise belongs to a family of elliptically symmetric distributions, including Gaussian, Student-$t$ and Laplace laws, then the filter error stabilizes as predicted by the theory. The results provide a theoretical basis for quantifying the robustness of nonlinear filters under random observations and misspecified dynamics.
