Maximum Likelihood Estimation of Functionals of Discrete Distributions

Jiantao Jiao, Kartik Venkat, Yanjun Han, Tsachy Weissman

Research output: Contribution to journalArticlepeer-review


We consider the problem of estimating functionals of discrete distributions, and focus on a tight (up to universal multiplicative constants for each specific functional) nonasymptotic analysis of the worst case squared error risk of widely used estimators. We apply concentration inequalities to analyze the random fluctuation of these estimators around their expectations and the theory of approximation using positive linear operators to analyze the deviation of their expectations from the true functional, namely their bias. We explicitly characterize the worst case squared error risk incurred by the maximum likelihood estimator (MLE) in estimating the Shannon entropy H(P) = σSi=1-pi ln pi, and the power sum Fα (P) = σSi=1 piαα >0 , up to universal multiplicative constants for each fixed functional, for any alphabet size ∞ and sample size n for which the risk may vanish. As a corollary, for Shannon entropy estimation, we show that it is necessary and sufficient to have n ≥ S1/a observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider n ≥ s1/aα samples for the MLE to consistently estimate Fα (P), 0<α <1. The minimax rate-optimal estimators for both problems require1. S and S1/aαln Ssamples, which implies that the MLE has a strictly sub-optimal sample complexity. When 1<α <3/2 , we show that the worst case squared error rate of convergence for the MLE is n-2α-1) for infinite alphabet size, while the minimax squared error rate is (nln n)-2α-1). When α ≥3/2 , the MLE achieves the minimax optimal rate n-1 regardless of the alphabet size. As an application of the general theory, we analyze the Dirichlet prior smoothing techniques for Shannon entropy estimation. In this context, one approach is to plug-in the Dirichlet prior smoothed distribution into the entropy functional, while the other one is to calculate the Bayes estimator for entropy under the Dirichlet prior for squared error, which is the conditional expectation. We show that in general such estimators do not improve over the maximum likelihood estimator. No matter how we tune the parameters in the Dirichlet prior, this approach cannot achieve the minimax rates in entropy estimation. The performance of the minimax rate-optimal estimator with n samples is essentially at least as good as that of Dirichlet smoothed entropy estimators with n\ln n samples.

Original languageEnglish (US)
Article number7997814
Pages (from-to)6774-6798
Number of pages25
JournalIEEE Transactions on Information Theory
Issue number10
StatePublished - Oct 2017


  • Dirichlet prior smoothing
  • Entropy estimation
  • Rényi entropy
  • approximation theory
  • approximation using positive linear operators
  • high dimensional statistics
  • maximum likelihood estimator

ASJC Scopus subject areas

  • Information Systems
  • Computer Science Applications
  • Library and Information Sciences


Dive into the research topics of 'Maximum Likelihood Estimation of Functionals of Discrete Distributions'. Together they form a unique fingerprint.

Cite this