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

Abstract

We introduce a novel hierarchical Bayesian permutation entropy (PermEn) estimator designed to improve biomedical time series entropy assessments, especially for short signals. Unlike existing methods requiring a substantial number of observations or which impose restrictive priors, our non-centered, Wasserstein optimized hierarchical approach enables efficient MCMC inference and a broader range of PermEn priors. Evaluations on synthetic and secondary benchmark data demonstrate superior performance over the current state-of-the-art, including 13.33-63.67% lower estimation error, 8.16-47.77% lower posterior variance, and 47-60.83% lower prior construction error ($p \leq 2.42 \times 10^{-10}$). Applied to cardiopulmonary exercise test oxygen uptake signals, we reveal a previously unreported 1.55% (95% credible interval: [0.62%, 2.52%]) entropy difference between obese and lean subjects that diminishes as exercise capacity increases. For individuals capable of completing at least 7.5 minutes of testing, the 95% credible interval contained zero, suggesting potential insights into physiological complexity, exercise tolerance, and obesity. Our estimator refines biomedical signal PermEn estimation and underscores entropy’s potential value as a health biomarker, opening avenues for further physiological and biomedical exploration.