  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.
