  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.
