Methods, systems, and apparatus, including computer programs encoded on computer storage media, for a neural network system comprising one or more gated linear networks. A system includes: one or more gated linear networks, wherein each gated linear network corresponds to a respective data value in an output data sample and is configured to generate a network probability output that defines a probability distribution over possible values for the corresponding data value, wherein each gated linear network comprises a plurality of layers, wherein the plurality of layers comprises a plurality of gated linear layers, wherein each gated linear layer has one or more nodes, and wherein each node is configured to: receive a plurality of inputs, receive side information for the node; combine the plurality of inputs according to a set of weights defined by the side information, and generate and output a node probability output for the corresponding data value.