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Bridging Bayesian and Information-Theoretic Approaches in Earth and Environmental System Modeling: Theory and Practical Applications

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Dr. Anneli Guthke

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Stuttgart Center for Simulation Science, Cluster of Excellence EXC 2075, University of Stuttgart, 70569 Stuttgart, Germany
Interests: Bayesian model assessment; information-theoretic concepts; model diagnostics; hybrid modeling; model choice uncertainty; data-integrated modeling; surrogate modeling
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Dr. Jasper A. Vrugt

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Department of Civil and Environmental Engineering, University of California Irvine, Irvine, CA 92697, USA
Interests: complex systems; data analysis; model evaluation; model diagnostics, probabilistic evaluation, uncertainty quantification; Bayesian estimation; statistical inference; information theory, sensitivity analysis

Dr. Steven V. Weijs

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Guest Editor

Department of Civil Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada
Interests: information theory; algorithmic information theory; hydrology; monitoring network design; probabilistic forecasting; forecast evaluation

Special Issue Information

Dear Colleagues,

Information theory has gained interest in the past decade in numerous scientific disciplines. Although Shannon information theory is rooted in probability theory, its connection with Bayesian theory is rarely explored, established or demonstrated in practical applications. We invite contributions that link information and Bayesian theory using theoretical examples or cross-cutting applications. We are especially interested in manuscripts with the following characteristics:

Manuscripts that demonstrate the practical usefulness and power of information-theoretic approaches for Earth and environmental system modeling, learning and prediction;
Manuscripts that highlight computational and/or implementation challenges;
Manuscripts that address theoretical limitations, pitfalls or shortcomings.

Topics may include, but are not limited to, the following: uncertainty quantification, hypothesis testing, (probabilistic) model evaluation, compression/learning, causal inference and decision-making under uncertainty.

Dr. Anneli Guthke
Dr. Jasper A. Vrugt
Dr. Steven V. Weijs
Guest Editors

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57 pages, 12419 KB

Open AccessArticle

The Learning Rate Is Not a Constant: Sandwich-Adjusted Markov Chain Monte Carlo Simulation

by Jasper A. Vrugt and Cees G. H. Diks

Entropy 2025, 27(10), 999; https://doi.org/10.3390/e27100999 - 25 Sep 2025

Abstract

A fundamental limitation of maximum likelihood and Bayesian methods under model misspecification is that the asymptotic covariance matrix of the pseudo-true parameter vector

θ_{*}

is not the inverse of the Fisher information, but rather the sandwich covariance matrix [...] Read more.

A fundamental limitation of maximum likelihood and Bayesian methods under model misspecification is that the asymptotic covariance matrix of the pseudo-true parameter vector

θ_{*}

is not the inverse of the Fisher information, but rather the sandwich covariance matrix

\frac{1}{n} A_{*}^{- 1} B_{*}^{} A_{*}^{- 1}

, where

A_{*}

and

B_{*}

are the sensitivity and variability matrices, respectively, evaluated at

θ_{*}

for training data record

ω_{1}, \dots, ω_{n}

. This paper makes three contributions. First, we review existing approaches to robust posterior sampling, including the open-faced sandwich adjustment and magnitude- and curvature-adjusted Markov chain Monte Carlo (MCMC) simulation. Second, we introduce a new sandwich-adjusted MCMC method. Unlike existing approaches that rely on arbitrary matrix square roots, eigendecompositions or a single scaling factor applied uniformly across the parameter space, our method employs a parameter-dependent learning rate

λ (θ)

that enables direction-specific tempering of the likelihood. This allows the sampler to capture directional asymmetries in the sandwich distribution, particularly under model misspecification or in small-sample regimes, and yields credible regions that remain valid when standard Bayesian inference underestimates uncertainty. Third, we propose information-theoretic diagnostics for quantifying model misspecification, including a strictly proper divergence score and scalar summaries based on the Frobenius norm, Earth mover’s distance, and the Herfindahl index. These principled diagnostics complement residual-based metrics for model evaluation by directly assessing the degree of misalignment between the sensitivity and variability matrices,

A_{*}

and

B_{*}

. Applications to two parametric distributions and a rainfall-runoff case study with the Xinanjiang watershed model show that conventional Bayesian methods systematically underestimate uncertainty, while the proposed method yields asymptotically valid and robust uncertainty estimates. Together, these findings advocate for sandwich-based adjustments in Bayesian practice and workflows. Full article

(This article belongs to the Special Issue Bridging Bayesian and Information-Theoretic Approaches in Earth and Environmental System Modeling: Theory and Practical Applications)

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Bridging Bayesian and Information-Theoretic Approaches in Earth and Environmental System Modeling: Theory and Practical Applications

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