Adaptive Estimation of Shannon Entropy

We consider estimating the Shannon entropy of a discrete distribution P from n i.i.d. samples. Recently, Jiao, Venkat, Han, and Weissman (JVHW), and Wu and Yang constructed approximation theoretic estimators that achieve the minimax L-2 rates in estimating entropy. Their estimators are consistent given n >> S/lnS samples, where S is the support size, and it is the best possible sample complexity. In contrast, the Maximum Likelihood Estimator (MLE), which is the empirical entropy, requires n >> S samples. In the present paper we significantly refine the minimax results of existing work. To alleviate the pessimism of minimaxity, we adopt the adaptive estimation framework, and show that the JVHW estimator is an adaptive estimator, i.e., it achieves the minimax rates simultaneously over a nested sequence of subsets of distributions P, without knowing the support size S or which subset P lies in. We also characterize the maximum risk of the MLE over this nested sequence, and show, for every subset in the sequence, that the performance of the minimax rate-optimal estimator with n samples is essentially that of the MLE with n ln n samples, thereby further substantiating the generality of "effective sample size enlargement" phenomenon discovered by Jiao, Venkat, Han, and Weissman. We provide a "pointwise" explanation of the sample size enlargement phenomenon, which states that for sufficiently small probabilities, the bias function of the JVHW estimator with n samples is nearly that of the MLE with n ln n samples.