Bayesian filtering serves as the mainstream framework of state estimation in dynamic systems. Its standard version utilizes total probability rule and Bayes' law alternatively, where how to define and compute conditional probability is critical to state distribution inference. Previously, the conditional probability is assumed to be exactly known, which represents a measure of the occurrence probability of one event, given the second event. In this paper, we find that by adding an additional event that stipulates an inequality condition, we can transform the conditional probability into a special integration that is analogous to convolution. Based on this transformation, we show that both transition probability and output probability can be generalized to convolutional forms, resulting in a more general filtering framework that we call convolutional Bayesian filtering. This new framework encompasses standard Bayesian filtering as a special case when the distance metric of the inequality condition is selected as Dirac delta function. It also allows for a more nuanced consideration of model mismatch by choosing different types of inequality conditions. For instance, when the distance metric is defined in a distributional sense, the transition probability and output probability can be approximated by simply rescaling them into fractional powers. Under this framework, a robust version of Kalman filter can be constructed by only altering the noise covariance matrix, while maintaining the conjugate nature of Gaussian distributions. Finally, we exemplify the effectiveness of our approach by reshaping classic filtering algorithms into convolutional versions, including Kalman filter, extended Kalman filter, unscented Kalman filter and particle filter.

翻译：贝叶斯滤波作为动态系统中状态估计的主流框架，其标准版本交替使用全概率法则与贝叶斯定律，其中条件概率的定义与计算对状态分布推理至关重要。传统方法假设条件概率精确已知，该概率表示在给定第二事件条件下第一事件的发生概率。本文发现，通过引入一个规定不等式条件的附加事件，可将条件概率转化为类似于卷积的特殊积分形式。基于这一变换，我们证明转移概率与输出概率均可泛化为卷积形式，由此提出更通用的滤波框架——卷积贝叶斯滤波。当不等式条件中的距离度量取狄拉克δ函数时，该新框架退化为标准贝叶斯滤波的特例。通过选择不同不等式条件，框架还能更精细地处理模型失配问题。例如，当距离度量以分布形式定义时，转移概率与输出概率可通过缩放为分数幂进行近似。在此框架下，仅需修改噪声协方差矩阵即可构建鲁棒卡尔曼滤波器，同时保持高斯分布的共轭特性。最后，我们通过将经典滤波算法（包括卡尔曼滤波、扩展卡尔曼滤波、无迹卡尔曼滤波及粒子滤波）重构为卷积版本，验证了该方法的有效性。