  From thermodynamic origins, the concept of entropy has expanded to a range of
statistical measures of uncertainty, which may still be thermodynamically
significant. However, laboratory measurements of entropy continue to rely on
direct measurements of heat. New technologies that can map out myriads of
microscopic degrees of freedom suggest direct determination of configurational
entropy by counting in systems where it is thermodynamically inaccessible, such
as granular and colloidal materials, proteins and lithographically fabricated
nanometre-scale arrays. Here, we demonstrate a conditional-probability
technique to calculate entropy densities of translation-invariant states on
lattices using limited configuration data on small clusters, and apply it to
arrays of interacting nanometre-scale magnetic islands (artificial spin ice).
Models for statistically disordered systems can be assessed by applying the
method to relative entropy densities. For artificial spin ice, this analysis
shows that nearest-neighbour correlations drive longer-range ones.
