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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) and Kernel Density (KD) methods. 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. And, whereas BC and KD each require optimal tuning of a hyper-parameter whose value varies with sample size and shape of the pdf, QS only requires specification of the number of quantiles to be used. Results indicate, for the class of distributions tested, that the optimal number of quantiles 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. This provides a clear advantage over BC and KD since hyper-parameter tuning is not required. Further, unlike KD, there is no need to select an appropriate kernel-type, and so QS is applicable to pdfs of arbitrary shape, including those with discontinuous slope and/or magnitude. 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 low even for samples of as few as 100 data points; in contrast, for KD the small sample bias can be as large as −10% and for BC as large as −50%. 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. 

The basis for all knowledge is “information” that we compile about the world, expressed through models that support understanding, prediction, and decision making. This overview paper provides a contextual basis for the four papers that make up the “debate series” compiled under the above title. We briefly introduce Information Theory, discuss how “information” can be considered to be both a “physical” quantity and a “probabilistic” basis for representing incompleteness in knowledge, discuss the core motivation for this debate series, and briefly summarize the major arguments advanced by each of the debate papers. Our purpose is to facilitate an understanding of how these papers are related and how they approach the debate series question from different perspectives, while pointing to future directions for research. Finally, we invite further discourse and debate to advance the understanding and prediction of natural system dynamics using Information Theory, including the assessment of its limitations and complementarity to existing physics and machine learning approaches. Ultimately, our goal is to press for the development of philosophical and methodological advances that will enable the Earth science community to address some of the compelling unsolved problems in our field.