Empirical estimation of entropy functionals with confidence

Abstract

This paper introduces a class of k-nearest neighbor ($k$-NN) estimators called bipartite plug-in (BPI) estimators for estimating integrals of non-linear functions of a probability density, such as Shannon entropy and Rényi entropy. The density is assumed to be smooth, have bounded support, and be uniformly bounded from below on this set. Unlike previous $k$-NN estimators of non-linear density functionals, the proposed estimator uses data-splitting and boundary correction to achieve lower mean square error. Specifically, we assume that $T$ i.i.d. samples $X_i\in {\mathbb{R}}^d$ from the density are split into two pieces of cardinality $M$ and $N$ respectively, with $M$ samples used for computing a k-nearest-neighbor density estimate and the remaining $N$ samples used for empirical estimation of the integral of the density functional. By studying the statistical properties of k-NN balls, explicit rates for the bias and variance of the BPI estimator are derived in terms of the sample size, the dimension of the samples and the underlying probability distribution. Based on these results, it is possible to specify optimal choice of tuning parameters $M/T$, $k$ for maximizing the rate of decrease of the mean square error (MSE). The resultant optimized BPI estimator converges faster and achieves lower mean squared error than previous $k$-NN entropy estimators. In addition, a central limit theorem is established for the BPI estimator that allows us to specify tight asymptotic confidence intervals.

Version 3 changes : Additional results on bias correction factors for specifically estimating Shannon and Renyi entropy in Section 5

References in corpus (1)

• A class of Rényi information estimators for multidimensional densities

Cited by in corpus (5)

• Nonparametric Divergence Estimation with Applications to Machine Learning on Distributions
• Analysis of k-Nearest Neighbor Distances with Application to Entropy Estimation
• Kernels on Sample Sets via Nonparametric Divergence Estimates
• Nonparanormal Information Estimation
• co-BPM: a Bayesian Model for Divergence Estimation