Methods, systems, and apparatus, including computer programs encoded on a computer storage medium, for training an encoder neural network, a decoder neural network, and a prior neural network, and using the trained networks for generative modeling, data compression, and data decompression. In one aspect, a method comprises: providing a given observation as input to the encoder neural network to generate parameters of an encoding probability distribution; determining an updated code for the given observation; selecting a code that is assigned to an additional observation; providing the code assigned to the additional observation as input to the prior neural network to generate parameters of a prior probability distribution; sampling latent variables from the encoding probability distribution; providing the latent variables as input to the decoder neural network to generate parameters of an observation probability distribution; and determining gradients of a loss function.