Computing Accurate Probabilistic Estimates of One-D Entropy from Equiprobable Random Samples

TL;DR

This paper introduces a simple Quantile Spacing method for accurately estimating one-dimensional entropy from equiprobable samples, outperforming the traditional Bin-Counting approach by avoiding hyperparameter tuning and providing reliable uncertainty estimates.

Contribution

The paper presents the QS method for entropy estimation, which uses quantile-based intervals and is less sensitive to distributional assumptions than BC, with fixed optimal quantile count relative to sample size.

Findings

QS method achieves less than 1% bias across tested distributions.

Optimal number of quantiles is approximately 25-35% of sample size.

Bootstrapping accurately captures the entropy estimate uncertainty.

Abstract

We develop a simple Quantile Spacing (QS) method for accurate probabilistic estimation of one-dimensional entropy from equiprobable random samples, and compare it with the popular Bin-Counting (BC) method. In contrast to BC, which uses equal-width bins with varying probability mass, the QS method uses estimates of the quantiles that divide the support of the data generating probability density function (pdf) into equal-probability-mass intervals. Whereas BC requires optimal tuning of a bin-width hyper-parameter whose value varies with sample size and shape of the pdf, QS requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantile-spacings is a fixed fraction of the sample size (empirically determined to be ~0.25-0.35), and that this value is relatively insensitive to distributional form or sample size, providing a clear advantage over BC since hyperparameter tuning is not required. Bootstrapping is used to approximate the sampling variability distribution of the resulting entropy estimate, and is shown to accurately reflect the true uncertainty. For the four distributional forms studied (Gaussian, Log-Normal, Exponential and Bimodal Gaussian Mixture), expected estimation bias is less than 1% and uncertainty is relatively low even for very small sample sizes. We speculate that estimating quantile locations, rather than bin-probabilities, results in more efficient use of the information in the data to approximate the underlying shape of an unknown data generating pdf.