Federated learning describes the distributed training of models across multiple clients while keeping the data private on-device. In this work, we formalize the server-orchestrated federated learning process as a hierarchical latent variable model where the server provides the parameters of a prior distribution over the client-specific model parameters. We then show that with simple Gaussian priors and a hard version of the well known Expectation-Maximization (EM) algorithm, learning in such a model corresponds to FedAvg, the most popular algorithm for this federated learning setting. This perspective on federated learning unifies several recent works in the field and opens up the possibility for extensions through different choices in the hierarchical model. Based on this view, we further propose a variant of the hierarchical model that employs prior distributions to promote sparsity. By using the hard-EM algorithm for learning, we obtain FedSparse, a procedure that can learn sparse neural networks in the federated learning setting. FedSparse reduces communication costs from client to server and vice-versa, as well as the computational costs for inference with the sparsified network – both of which are of great practical importance in federated learning.