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Abstract This paper extends the application of quantile-based Bayesian inference to probability distributions defined in terms of quantiles of observable quantities. Quantile-parameterized distributions are characterized by high shape flexibility and parameter interpretability, making them useful for eliciting information about observables. To encode uncertainty in the quantiles elicited from experts, we propose a Bayesian model based on the metalog distribution and a variant of the Dirichlet prior. We discuss the resulting hybrid expert elicitation protocol, which aims to characterize uncertainty in parameters by asking questions about observable quantities. We also compare and contrast this approach with parametric and predictive elicitation methods.
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Computing Accurate Probabilistic Estimates of One-D Entropy from Equiprobable Random Samples
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.
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- Award ID(s):
- 1740858
- PAR ID:
- 10294762
- Date Published:
- Journal Name:
- Entropy
- Volume:
- 23
- Issue:
- 6
- ISSN:
- 1099-4300
- Page Range / eLocation ID:
- 740
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/e23060740
