We develop a simple and elegant method for lossless compression using latent variable models, which we call `bits back with asymmetric numeral systems' (BB-ANS). The method involves interleaving encode and decode steps, and achieves an optimal rate when compressing batches of data. We demonstrate it rstly on the MNIST test set, showing that state-of-the-art lossless compression is possible using a small variational autoencoder (VAE) model. We then make use of a novel empirical insight, that fully convolutional generative models, trained on small images, are able to generalize to images of arbitrary size, and extend BB-ANS to hierarchical latent variable models, enabling state-of-the-art lossless compression of full-size colour images from the ImageNet dataset. We describe `Craystack', a modular software framework which we have developed for rapid prototyping of compression using deep generative models
