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Entropy
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Bayesian Hierarchical Models with Applications
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Bayesian Hierarchical Models with Applications

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: 30 January 2026 | Viewed by 1165

Special Issue Editors

Prof. Dr. Maura Mezzetti

E-Mail Website
Guest Editor
Department of Economics and Finance, University of Rome Tor Vergata, 00133 Rome, Italy
Interests: Bayesian hierarchical model research evaluation; socioeconomic disparities and cancer burden; gender gap in academia
Prof. Dr. Ilia Negri

E-Mail Website
Guest Editor
Department of Economics, Statistics and Finance “Giovanni Anania”—DESF, University of Calabria, Via P. Bucci, Cubo 0C, 87036 Rende, CS, Italy
Interests: inference for dynamical systems; change point problems; statistical methods for gender health

Special Issue Information

Dear Colleagues,

Hierarchical Bayesian models have become increasingly widespread thanks to the development of numerous supporting software, starting from BUGS, JAGS, and Stan, all of which have comfortable interface. The interesting aspect lies in how the model is defined, the prior distributions, and the choice of levels in the hierarchical structure. On one hand, their simplicity in interpretation, and on the other, the ease of implementation, make these models highly versatile. They are, in fact, applied across a wide range of fields, from epidemiology to social and environmental sciences.

The use of multilevel models, with particular attention to random effect models, is highly important. Furthermore, a discussion on their definition and, more importantly, how to interpret the posterior estimations would greatly benefit the statistical community.

Prof. Dr. Maura Mezzetti
Prof. Dr. Ilia Negri
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Bayesian inference
  • MCMC
  • multilevel model
  • random effects
  • model comparison
  • posterior distributions
  • Stan

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Published Papers (2 papers)

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Research

24 pages, 2253 KiB  
Article
Modeling Spatial Data with Heteroscedasticity Using PLVCSAR Model: A Bayesian Quantile Regression Approach
by Rongshang Chen and Zhiyong Chen
Entropy 2025, 27(7), 715; https://doi.org/10.3390/e27070715 - 1 Jul 2025
Viewed by 289
Abstract
Spatial data not only enables smart cities to visualize, analyze, and interpret data related to location and space, but also helps departments make more informed decisions. We apply a Bayesian quantile regression (BQR) of the partially linear varying coefficient spatial autoregressive (PLVCSAR) model [...] Read more.
Spatial data not only enables smart cities to visualize, analyze, and interpret data related to location and space, but also helps departments make more informed decisions. We apply a Bayesian quantile regression (BQR) of the partially linear varying coefficient spatial autoregressive (PLVCSAR) model for spatial data to improve the prediction of performance. It can be used to capture the response of covariates to linear and nonlinear effects at different quantile points. Through an approximation of the nonparametric functions with free-knot splines, we develop a Bayesian sampling approach that can be applied by the Markov chain Monte Carlo (MCMC) approach and design an efficient Metropolis–Hastings within the Gibbs sampling algorithm to explore the joint posterior distributions. Computational efficiency is achieved through a modified reversible-jump MCMC algorithm incorporating adaptive movement steps to accelerate chain convergence. The simulation results demonstrate that our estimator exhibits robustness to alternative spatial weight matrices and outperforms both quantile regression (QR) and instrumental variable quantile regression (IVQR) in a finite sample at different quantiles. The effectiveness of the proposed model and estimation method is demonstrated by the use of real data from the Boston median house price. Full article
(This article belongs to the Special Issue Bayesian Hierarchical Models with Applications)
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35 pages, 5260 KiB  
Article
Physics-Informed Neural Networks with Unknown Partial Differential Equations: An Application in Multivariate Time Series
by Seyedeh Azadeh Fallah Mortezanejad, Ruochen Wang and Ali Mohammad-Djafari
Entropy 2025, 27(7), 682; https://doi.org/10.3390/e27070682 - 26 Jun 2025
Viewed by 579
Abstract
A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: How can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? [...] Read more.
A significant advancement in Neural Network (NN) research is the integration of domain-specific knowledge through custom loss functions. This approach addresses a crucial challenge: How can models utilize physics or mathematical principles to enhance predictions when dealing with sparse, noisy, or incomplete data? Physics-Informed Neural Networks (PINNs) put this idea into practice by incorporating a forward model, such as Partial Differential Equations (PDEs), as soft constraints. This guidance helps the networks find solutions that align with established laws. Recently, researchers have expanded this framework to include Bayesian NNs (BNNs) which allow for uncertainty quantification. However, what happens when the governing equations of a system are not completely known? In this work, we introduce methods to automatically select PDEs from historical data in a parametric family. We then integrate these learned equations into three different modeling approaches: PINNs, Bayesian-PINNs (B-PINNs), and Physical-Informed Bayesian Linear Regression (PI-BLR). To assess these frameworks, we evaluate them on a real-world Multivariate Time Series (MTS) dataset related to electrical power energy management. We compare their effectiveness in forecasting future states under different scenarios: with and without PDE constraints and accuracy considerations. This research aims to bridge the gap between data-driven discovery and physics-guided learning, providing valuable insights for practical applications. Full article
(This article belongs to the Special Issue Bayesian Hierarchical Models with Applications)
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