  The network reconstruction task aims to estimate a complex system's structure
from various data sources such as time series, snapshots, or interaction
counts. Recent work has examined this problem in networks whose relationships
involve precisely two entities-the pairwise case. Here we investigate the
general problem of reconstructing a network in which higher-order interactions
are also present. We study a minimal example of this problem, focusing on the
case of hypergraphs with interactions between pairs and triplets of vertices,
measured imperfectly and indirectly. We derive a
Metropolis-Hastings-within-Gibbs algorithm for this model and use the
algorithms to highlight the unique challenges that come with estimating
higher-order models. We show that this approach tends to reconstruct empirical
and synthetic networks more accurately than an equivalent graph model without
higher-order interactions.
