A novel reference prior for Gaussian hierarchical models with intrinsic conditional autoregressive random effects

2026-04-13

AI Summary

This study addresses the computational burden of Bayesian variable selection in Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) random effects, where the conventional approach incurs a complexity of O(n³2ᵏ). To overcome this challenge, the authors propose a novel reference prior that requires only a single spectral decomposition of an n-dimensional matrix and is applicable across all candidate models. This innovation reduces the computational complexity to O(n³) while preserving the variable selection performance of the original method. Empirical evaluations on simulated data with ten regression variables and a real-world analysis of U.S. county-level income demonstrate over a 1,000-fold acceleration in computation, achieving a remarkable balance between computational efficiency and statistical accuracy.

Technology Category

• Reasoning under Uncertainty: Graphical Models
• Machine Learning: Bayesian Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

We develop a novel reference prior for Gaussian hierarchical models with intrinsic conditional autoregressive (ICAR) random effects. This is particularly important in the context of objective Bayes variable selection with sample size $n$ and $k$ regressors. In this context, a previously published reference prior requires the computation of spectral decompositions of two $n$-dimensional matrices for each model under consideration. As a consequence, for variable selection the computational cost of this previous reference prior grows as $O(n^3 2^k)$. In contrast, our novel reference prior requires the computation of the spectral decomposition of one $n$-dimensional matrix that can be used for all models under consideration. Thus, the computational cost of our novel reference prior grows much slower as $O(n^3)$. Hence, computational savings can be substantial, e.g. in a problem with 10 regressors, when compared to the previously published reference prior, computations based on our novel reference prior are more than 1000 times faster. We provide a proof of the equivalence of the two priors. A simulation study shows that, while both reference priors provide equivalent variable selection results, for large sample sizes computations based on our novel prior are several orders of magnitude faster. Finally, the utility of our novel reference prior is illustrated with a spatial regression study of county-level median household income on socio-economic regressors for 3108 counties in the contiguous United States.

Problem

• reference prior
• Gaussian hierarchical models
• intrinsic conditional autoregressive
• variable selection
• computational cost

Innovation

• reference prior
• Gaussian hierarchical model
• intrinsic conditional autoregressive
• variable selection
• computational efficiency