We propose a new approach for propagating stable probability distributions through neural networks. Our method is based on local linearization, which we show to be an optimal approximation in terms of total variation distance for the ReLU non-linearity. This allows propagating Gaussian and Cauchy input uncertainties through neural networks to quantify their output uncertainties. To demonstrate the utility of propagating distributions, we apply the proposed method to predicting calibrated confidence intervals and selective prediction on out-of-distribution data. The results demonstrate a broad applicability of propagating distributions and show the advantages of our method over other approaches such as moment matching.

翻译：我们提出一种新方法，用于通过神经网络传播稳定概率分布。该方法基于局部线性化，我们证明对于ReLU非线性激活函数，该线性化在总变差距离意义下是最优近似。这使得能够通过神经网络传播高斯和柯西输入不确定性，从而量化其输出不确定性。为展示分布传播的实用性，我们将所提方法应用于预测校准置信区间以及对分布外数据的选择性预测。结果表明分布传播具有广泛适用性，并展示了我们的方法相较于矩匹配等其他方法的优势。