Title: Efficient Certification for Probabilistic Robustness. Abstract: Recent developments on the robustness of neural networks have primarily emphasized the notion of worst-case adversarial robustness in both verification and robust training. However, often looser constraints are needed and some margin of error is allowed. We instead consider the task of probabilistic robustness, which assumes the input follows a known probabilistic distribution and seeks to bound the probability of a given network failing against the input. We focus on developing an efficient robustness verification algorithm by extending a bound-propagation-based approach. Our proposed algorithm improves upon the robustness certificate of this algorithm by up to $8\times$ while with no additional computational cost. In addition, we perform a case study on incorporating the probabilistic robustness verification during training for the first time.