Title: Fast Predictive Uncertainty for Classification with Bayesian Deep Networks. Abstract: In Bayesian Deep Learning, distributions over the output of classification neural networks are approximated by first constructing a Gaussian distribution over the weights, then sampling from it to receive a distribution over the categorical output distribution. This is costly. We reconsider old work to construct a Dirichlet approximation of this output distribution, which yields an analytic map between Gaussian distributions in logit space and Dirichlet distributions (the conjugate prior to the categorical) in the output space. We argue that the resulting Dirichlet distribution has theoretical and practical advantages, in particular, more efficient computation of the uncertainty estimate, scaling to large datasets and networks like ImageNet and DenseNet. We demonstrate the use of this Dirichlet approximation by using it to construct a lightweight uncertainty-aware output ranking for the ImageNet setup.