Title: Generative Posterior Networks for Approximately Bayesian Epistemic Uncertainty Estimation. Abstract: Ensembles of neural networks are often used to estimate epistemic uncertainty in high-dimensional problems because of their scalability and ease of use. These methods, however, are expensive to sample from as each sample requires a new neural network to be trained from scratch. We propose a new method, Generative Posterior Networks (GPNs), a generative model that, given a prior distribution over functions, approximates the posterior distribution directly by regularizing the network towards samples from the prior. This allows our method to quickly sample from the posterior and construct confidence bounds. We prove theoretically that our method indeed approximates the Bayesian posterior and show empirically that it improves epistemic uncertainty estimation over competing methods.