Uncertainty Quantification via Stable Distribution Propagation

Felix Petersen, Aashwin Ananda Mishra, Hilde Kuehne, Christian Borgelt, Oliver Deussen, Mikhail Yurochkin

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Keywords: propagating distributions, uncertainty, uncertainties, aleatoric, epistemic, moment matching, total variation, sampling-free, deterministic, variational inference, propagation, probabilistic neural networks, variance propagation, Cauchy, Cauchy distribution, Gaussian, analytical, data uncertainty, alpha stable

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Abstract: 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.

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Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

Submission Number: 4440