Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2604.23527

Using Statistical Mechanics to Improve Real-World Bayesian Inference: A New Method Combining Tempered Posteriors and Wang-Landau Sampling

Alfred C. K. Farris
View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a simple method to obtain optimal posterior distributions and improve the quality of Bayesian inference with reduced human and computational effort. Bayes' Theorem is reformulated in the language of statistical mechanics, wherein an improved posterior -- referred to as a tempered posterior -- is defined analogously to a canonical probability distribution at temperature $τ$. Wang-Landau sampling is used to obtain the density of states of the posterior probability, and signals analogous to those of phase transitions are extracted from a single simulation. In addition, the transition temperature is easily identified, providing the tempered posterior with optimal predictive performance. We demonstrate the efficacy of the method on a real-world problem in materials science (equation of state modeling) with messy data, a high-dimensional and correlated input parameter space, and "frustration" among model outputs.

Explore related subjects

stat.MEphysics.comp-phphysics.data-anstat.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Alfred C. K. Farris. 2026-04-26. Using Statistical Mechanics to Improve Real-World Bayesian Inference: A New Method Combining Tempered Posteriors and Wang-Landau Sampling. https://arxiv.org/abs/2604.23527

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bias-Correction for Privacy-Protected Spatial Autoregressive Models with Application to Restaurant Network Analysis

Spatial autoregressive (SAR) models and their extensions are important tools for studying network effects. However, with an increasing emphasis on data privacy, data providers often implement protection measures that render standard SAR models inapplicable. In this study, we introduce a privacy-protected SAR model that incorporates noise into both the response and covariates to meet privacy requirements. With noise present in both components, the traditional quasi-maximum likelihood estimator becomes difficult to compute because the likelihood function cannot be directly formulated. To bypass this hurdle, we begin with a pseudo-likelihood approach, initially omitting the noise in the covariates. A Newton-Raphson algorithm is then applied to compute the estimator; however, the estimator is biased. To address this, we propose a bias-corrected Newton-Raphson-type algorithm that simultaneously accounts for noise in both the response and covariates. We further show, under appropriate regularity conditions, that the resulting estimator is consistent and asymptotically normal. To further enhance computational efficiency, we also develop a bias-corrected least squares estimator. Several extensions are discussed, and the finite-sample performance of the proposed methods is evaluated through extensive simulations. We apply the proposed methodology to restaurant transaction data from a third-party payment platform. Our method identifies a statistically significant competitive network effect among restaurants and further reveals meaningful restaurant-customer interaction patterns.

stat.ME↗
arXiv2026

A variational framework for modal estimation

Multivariate mode estimation arises in many statistical problems such as inverse problems, multimodal sampling, and density-based clustering, but becomes challenging in moderate to high dimensions, especially when the underlying density is not directly evaluable. We introduce GERVE (Gibbs-measure Entropy-Regularized Variational Estimation), a sample-based method for estimating multivariate modes by approximating Gibbs distributions directly from samples, without estimating or evaluating the density. GERVE uses Gaussian-mixture variational annealing and natural-gradient optimization, producing a mixture concentrated in high-density regions whose component responsibilities also provide a clustering of the observations. We prove theoretical guarantees in two regimes: as the Gibbs temperature goes to zero, the optimal variational mixture concentrates around the global modes of the population density; at fixed positive temperature, we prove existence, consistency, and asymptotic normality of empirical maximizers and propose a bootstrap procedure for uncertainty quantification. Simulations and a real-data experiment show that GERVE accurately recovers modes and produces meaningful clusters.

stat.ME↗
arXiv2026

Objective Model Prior Probabilities in Variable Selection

For many years it was routine to use equal model prior probabilities in Bayesian model uncertainty analysis. At least twenty years ago it became clear that this was problematic, leading to support of much too large models in the increasingly huge model spaces being considered in genomics and other fields. A popular replacement was to adopt a suggestion of Harold Jeffreys for the variable selection problem in which a total of $k$ possible variables are being considered for inclusion in the model: give the collection of all models containing $d$ variables ($d = 0, . . . , k$) prior probability $1/(k + 1)$ and then divide this prior probability equally among the models in the collection. Many other choices of model prior probabilities that impose severe parsimony have also been introduced. We begin by reviewing the problems with using equal model prior probabilities and then discuss some serious problems with the Jeffreys choice. Finally, we introduce and study a number of objective alternative choices of model prior probabilities, from both numerical and theoretical perspectives.

stat.ME↗