  Uncertainty quantification of neural networks is critical to measuring the
reliability and robustness of deep learning systems. However, this often
involves costly or inaccurate sampling methods and approximations. This paper
presents a sample-free moment propagation technique that propagates mean
vectors and covariance matrices across a network to accurately characterize the
input-output distributions of neural networks. A key enabler of our technique
is an analytic solution for the covariance of random variables passed through
nonlinear activation functions, such as Heaviside, ReLU, and GELU. The wide
applicability and merits of the proposed technique are shown in experiments
analyzing the input-output distributions of trained neural networks and
training Bayesian neural networks.
