Rad, Najmeh NakhaeiArashi, MohammadBekker, Andriette, 1958-Millard, Sollie M.2025-07-162025-07-162025-12Rad, N.N., Arashi, M., Bekker, A. & Millard, S. 2025, 'Bayesian nonparametric estimation of differential entropy for toroidal data', Applied Mathematical Modelling, vol. 148, art. 116241, pp. 1-15, doi : 10.1016/j.apm.2025.116241.0307-904X (print)1872-8480 (online)10.1016/j.apm.2025.116241http://hdl.handle.net/2263/103391DATA AVAILABILITY : The wind data is available upon request, and the protein data is available at http://scop.mrc-lmb.cam.ac.uk/scop/. .Entropy is a widely used information-theoretic measure; however, the estimation of entropy for observations of a periodic nature has not received much attention thus far. In this paper, we implement a Bayesian approach to obtain nonparametric estimates of Shannon entropy for toroidal data. This paves the way for its use in protein structure validation through an approach based on information theory and the distribution of backbone dihedral angles in the 3D structure of proteins. In addition, the kernel density estimation proposed in this paper can be applied alongside available parametric models for modeling toroidal observations. Simulation studies and the analysis of real datasets provide insights into this proposed method for protein structure validation. HIGHLIGHTS • Estimation of entropy for observations of a periodic nature specifically toroidal data. • A Bayesian nonparametric density estimator for toroidal data using a Dirichlet infinite mixture model. • Considering the possible dependencies between two circular variables. • An alternative information-theory based method for protein structure validation. • Diagnosing of the disordered pattern of wind direction in univariate case.en© 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).Bayesian nonparamertic inferenceDirichlet infinite mixture modelModified Gibbs samplingPlug-in estimatesShannon entropyArticle