  The profile of a sample is the multiset of its symbol frequencies. We show
that for samples of discrete distributions, profile entropy is a fundamental
measure unifying the concepts of estimation, inference, and compression.
Specifically, profile entropy a) determines the speed of estimating the
distribution relative to the best natural estimator; b) characterizes the rate
of inferring all symmetric properties compared with the best estimator over any
label-invariant distribution collection; c) serves as the limit of profile
compression, for which we derive optimal near-linear-time block and sequential
algorithms. To further our understanding of profile entropy, we investigate its
attributes, provide algorithms for approximating its value, and determine its
magnitude for numerous structural distribution families.
