Probabilistic models with discrete latent variables naturally capture datasets composed of discrete classes. However, they are difficult to train efficiently, since back propagation through discrete variables is generally not possible. We introduce a novel class of probabilistic models, comprising an undirected discrete component and a directed hierarchical continuous component, that can be trained efficiently using the variational autoencoder framework. The discrete component captures the distribution over the disconnected smooth manifolds induced by the continuous component. As a result, this class of models efficiently learns both the class of objects in an image, and their specific realization in pixels, from unsupervised data; and outperforms state-of-the-art methods on the permutation-invariant MNIST, OMNIGLOT, and Caltech-101 Silhouettes datasets.