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The profile of a sample is the multiset of its symbol frequencies. We showthat for samples of discrete distributions, profile entropy is a fundamentalmeasure unifying the concepts of estimation, inference, and compression.Specifically, profile entropy a) determines the speed of estimating thedistribution relative to the best natural estimator; b) characterizes the rateof inferring all symmetric properties compared with the best estimator over anylabel-invariant distribution collection; c) serves as the limit of profilecompression, for which we derive optimal near-linear-time block and sequentialalgorithms. To further our understanding of profile entropy, we investigate itsattributes, provide algorithms for approximating its value, and determine itsmagnitude for numerous structural distribution families.

Profile Entropy: A Fundamental Measure for the Learnability and Compressibility of Discrete Distributions