Fast Predictive Uncertainty for Classification with Bayesian Deep Networks

Marius Hobbhahn, Agustinus Kristiadi, Philipp Hennig

Keywords: Bayesian Deep Learning, Approximate Inference, Bayesian Inference, Fast Inference, Laplace Bridge

TL;DR: We propose and analyse a simple method to replace MC-integration in Bayesian Neural Networks in classification settings

Abstract:

In Bayesian Deep Learning, distributions over the output of classification neural networks are often approximated by first constructing a Gaussian distribution over the weights, then sampling from it to receive a distribution over the softmax outputs. This is costly. We reconsider old work (Laplace Bridge) to construct a Dirichlet approximation of this softmax output distribution, which yields an analytic map between Gaussian distributions in logit space and Dirichlet distributions (the conjugate prior to the Categorical distribution) in the output space.
Importantly, the vanilla Laplace Bridge comes with certain limitations. We analyze those and suggest a simple solution that compares favorably to other commonly used estimates of the softmax-Gaussian integral.
We demonstrate that the resulting Dirichlet distribution has multiple advantages, in particular, more efficient computation of the uncertainty estimate and scaling to large datasets and networks like ImageNet and DenseNet.
We further demonstrate the usefulness of this Dirichlet approximation by using it to construct a lightweight uncertainty-aware output ranking for ImageNet.