Maximum Likelihood Estimation of Functionals of Discrete Distributions

Abstract

We consider the problem of estimating functionals of discrete distributions, and focus on tight 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 \emph{bias}. We characterize the worst case squared error risk incurred by the Maximum Likelihood Estimator (MLE) in estimating the Shannon entropy $H(P) = \sum_{i = 1}^S -p_i \ln p_i$, and $F_\alpha(P) = \sum_{i = 1}^S p_i^\alpha,\alpha>0$, up to multiplicative constants, for any alphabet size $S\leq \infty$ 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 $n \gg S$ observations for the MLE to be consistent. In addition, we establish that it is necessary and sufficient to consider $n \gg S^{1/\alpha}$ samples for the MLE to consistently estimate $F_\alpha(P), 0<\alpha<1$. The minimax rate-optimal estimators for both problems require $S/\ln S$ and $S^{1/\alpha}/\ln S$ samples, which implies that the MLE has a strictly sub-optimal sample complexity. When $1<\alpha<3/2$, we show that the worst-case squared error rate of convergence for the MLE is $n^{-2(\alpha-1)}$ for infinite alphabet size, while the minimax squared error rate is $(n\ln n)^{-2(\alpha-1)}$. When $\alpha\geq 3/2$, 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. We show that no matter how we tune the parameters in the Dirichlet prior, this technique cannot achieve the minimax rates in entropy estimation.