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We study the problem of distributed filtering for state space models over networks, which aims to collaboratively estimate the states by a network of nodes. Most of existing works on this problem assume that both process and measurement noises are Gaussian and their covariances are known in advance. In some cases, this assumption breaks down and no longer holds. In this paper, we consider the case that both process and measurement noise covariances are unknown. A few related works have studied this problem. However, these works consider the situation of centralized processing, and they only derive smoothers which are not suitable for online processing in a network. Instead, this paper presents a novel robust Bayesian filter with unknown process and measurement noise covariances over sensor networks, which is distributed and online. A novel Bayesian model is formulated for a modified state space model. This Bayesian model is capable of dealing with outliers and heavy-tailed noises and improving the robustness of the filter to these non-Gaussian noises. Using this model, we first propose a novel centralized algorithm for the robust Bayesian filtering based on variational Bayesian methods. Then, we extend it to the distributed scenario based on the alternating direction method of multipliers (ADMM) technique. The proposed algorithm can simultaneously estimate the states and process/measurement noise covariances. Simulations on a target tracking problem are given to verify the effectiveness of the proposed model and algorithm. The results demonstrate that the proposed algorithm performs much better than the standard Kalman filter and as good as the corresponding centralized algorithm in the presence of outliers.