  Hypergraphs, encoding structured interactions among any number of system
units, have recently proven a successful tool to describe many real-world
biological and social networks. Here we propose a framework based on
statistical inference to characterize the structural organization of
hypergraphs. The method allows to infer missing hyperedges of any size in a
principled way, and to jointly detect overlapping communities in presence of
higher-order interactions. Furthermore, our model has an efficient numerical
implementation, and it runs faster than dyadic algorithms on pairwise records
projected from higher-order data. We apply our method to a variety of
real-world systems, showing strong performance in hyperedge prediction tasks,
detecting communities well aligned with the information carried by
interactions, and robustness against addition of noisy hyperedges. Our approach
illustrates the fundamental advantages of a hypergraph probabilistic model when
modeling relational systems with higher-order interactions.
