A General Framework for Symmetric Property Estimation

Symmetric property estimation is a fundamental and well studied problem in machine learning and statistics. In this problem, we are given n i.i.d samples from an unknown distribution 1 p and asked to estimate f(p), where f is a symmetric property (i.e. it does not depend on the labels of the symbols). Over the past few years, the computational and sample complexities for estimating many symmetric properties have been extensively studied. Estimators with optimal sample complexities have been obtained for several properties including entropy [VV11b, WY16a, JVHW15], distance to uniformity [VV11a, JHW16], and support [VV11b, WY15]. All aforementioned estimators were property specific and therefore, a natural question is to design a universal estimator. In [ADOS16], the authors showed that the distribution that maximizes the profile likelihood, i.e. the likelihood of the multiset of frequencies of elements in the sample, referred to as profile maximum likelihood (PML) distribution, can be used as a universal plugin estimator.