  We present a scheme for investigating arbitrary thermal observables in
spatially inhomogeneous equilibrium many-body systems. Extending the grand
canonical ensemble yields any given observable as an explicit hyper-density
functional. Associated local fluctuation profiles follow from an exact
hyper-Ornstein-Zernike equation. While the local compressibility and simple
observables permit analytic treatment, complex order parameters are accessible
via simulation-based supervised machine learning of neural hyper-direct
correlation functionals. We exemplify efficient and accurate neural predictions
for the cluster statistics of hard rods, square well rods, and hard spheres.
The theory allows to treat complex observables, as is impossible in standard
density functional theory.
