  Traditional Bayesian approaches for model uncertainty quantification rely on
notoriously difficult processes of marginalization over each network parameter
to estimate its probability density function (PDF). Our hypothesis is that
internal layer outputs of a trained neural network contain all of the
information related to both its mapping function (quantified by its weights) as
well as the input data distribution. We therefore propose a framework for
predictive uncertainty quantification of a trained neural network that
explicitly estimates the PDF of its raw prediction space (before activation),
p(y'|x,w), which we refer to as the model PDF, in a Gaussian reproducing kernel
Hilbert space (RKHS). The Gaussian RKHS provides a localized density estimate
of p(y'|x,w), which further enables us to utilize gradient based formulations
of quantum physics to decompose the model PDF in terms of multiple local
uncertainty moments that provide much greater resolution of the PDF than the
central moments characterized by Bayesian methods. This provides the framework
with a better ability to detect distributional shifts in test data away from
the training data PDF learned by the model. We evaluate the framework against
existing uncertainty quantification methods on benchmark datasets that have
been corrupted using common perturbation techniques. The kernel framework is
observed to provide model uncertainty estimates with much greater precision
based on the ability to detect model prediction errors.
