An Improved Bayesian Permutation Entropy Estimator with Wasserstein-Optimized Hierarchical Priors

Zachary Blanks, Donald E. Brown, Marc A. Adams, Siddhartha S. Angadi

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Abstract: We introduce a novel hierarchical Bayesian estimator for permutation entropy (PermEn), designed to improve the accuracy of entropy assessments of biomedical time series signal sets, particularly for short-duration signals. Unlike existing methods that require a substantial number of observations or impose restrictive priors, our approach uses a non-centered, Wasserstein distance optimized hierarchical prior, enabling efficient full Markov Chain Monte Carlo inference and a broader spectrum of PermEn priors. Comparative evaluations with synthetic and secondary benchmark data demonstrate our estimator's enhanced performance, including a significant reduction in estimation error (13.33-63.67\%), posterior variance (8.16-47.77\%), and reference prior distance error (47-60.83\%, $p \leq 2.42 \times 10^{-10}$) against current state-of-the-art methods. Applied to oxygen uptake signals from cardiopulmonary exercise testing, our method revealed a previously unreported entropy difference between obese and lean subjects (mean difference: 1.732\%; 94\% CI [2.34\%, 1.11\%], $p \leq \frac{1}{20000}$), with more precise credible intervals (16-24\% improvement). This entropy disparity becomes statistically non-significant in participants completing over 7.5 minutes of testing, suggesting potential insights into physiological complexity, exercise tolerance, and obesity. Our estimator thus not only refines the estimation of PermEn in biomedical signals but also underscores entropy's potential value as a health biomarker, opening avenues for further physiological and biomedical exploration.