We present a novel method for online inference of real-valued quantities on a large network from very sparse measurements. The target application is a large scale system, like e.g. a traffic network, where a small varying subset of the variables is observed, and predictions about the other variables have to be continuously updated. A key feature of our approach is the modeling of dependencies between the original variables through a latent binary Markov random field. This greatly simplifies both the model selection and its subsequent use. We introduce the mirror belief propagation algorithm, that performs fast inference in such a setting. The offline model estimation relies only on pairwise historical data and its complexity is linear w.r.t. the dataset size. Our method makes no assumptions about the joint and marginal distributions of the variables but is primarily designed with multimodal joint distributions in mind. Numerical experiments demonstrate both the applicability and scalability of the method in practice.