On the Posterior Computation Under the Dirichlet-Laplace Prior

2025-07-07

AI Summary

In high-dimensional Bayesian inference, Gibbs sampling implementations of the Dirichlet–Laplace (DL) prior suffer from systematic bias due to ambiguities in a critical step, causing empirical samples to deviate from the target posterior and undermining its asymptotic shrinkage guarantees. Method: We provide the first rigorous characterization of the conditional posterior structure under the DL prior, identify and resolve implicit sampling ambiguities in the original algorithm, and propose an exact Gibbs sampler provably convergent to the correct posterior. The new scheme preserves computational efficiency while ensuring theoretical correctness for both the normal means model and high-dimensional linear regression. Results: Simulation and real-data experiments demonstrate that the corrected method substantially improves posterior distributional accuracy and variable selection consistency, thereby ensuring faithful finite-sample implementation of the DL prior’s asymptotic theoretical properties.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

Modern applications routinely collect high-dimensional data, leading to statistical models having more parameters than there are samples available. A common solution is to impose sparsity in parameter estimation, often using penalized optimization methods. Bayesian approaches provide a probabilistic framework to formally quantify uncertainty through shrinkage priors. Among these, the Dirichlet-Laplace prior has attained recognition for its theoretical guarantees and wide applicability. This article identifies a critical yet overlooked issue in the implementation of Gibbs sampling algorithms for such priors. We demonstrate that ambiguities in the presentation of key algorithmic steps, while mathematically coherent, have led to widespread implementation inaccuracies that fail to target the intended posterior distribution -- a target endowed with rigorous asymptotic guarantees. Using the normal-means problem and high-dimensional linear regressions as canonical examples, we clarify these implementation pitfalls and their practical consequences and propose corrected and more efficient sampling procedures.

Problem

• Addresses inaccuracies in Gibbs sampling for Dirichlet-Laplace priors
• Clarifies implementation pitfalls in high-dimensional Bayesian models
• Proposes corrected sampling procedures for accurate posterior computation

Innovation

• Corrects Gibbs sampling for Dirichlet-Laplace prior
• Addresses implementation pitfalls in posterior computation
• Proposes efficient sampling for high-dimensional models