prior specification

This paper addresses computational and modeling bottlenecks in Bayesian regularization for multivariate statistical models. Methodologically, it introduces a unified Bayesian framework integrating parameter quantization (rounding) and approximate inference: shrinkage estimators—such as those shrinking toward a common mean—are rigorously interpreted as posterior means under specific priors, and an efficient approximate Bayesian algorithm is developed to balance accuracy and scalability. Theoretically, it establishes a rigorous connection between Bayesian inference and numerical approximation, providing the first unified Bayesian interpretation of classical shrinkage strategies. Applicationally, the framework is extended to regularized Linear Discriminant Analysis (LDA), yielding substantial improvements in classification stability and parameter estimation accuracy on both synthetic and real-world datasets. The approach thus bridges theoretical rigor with practical efficiency in high-dimensional multivariate inference.

This paper addresses the unsolved Bayesian dynamic borrowing problem—how to construct an informative prior when only a single external study is available. We propose a Meta-analytic-predictive (MAP) prior method tailored for the single-study setting. Within a normal–normal hierarchical modeling framework, our approach integrates shrinkage estimation with dynamic borrowing principles, providing the first systematic formalization of MAP prior construction—including explicit specification of key prior assumptions and their sensitivity. The method delivers robust parameter shrinkage while rigorously quantifying uncertainty. Evaluated on clinical medicine case studies, it demonstrates feasibility, interpretability, and improved statistical power, thereby bridging a critical theoretical and practical gap in Bayesian evidence synthesis under sparse-data conditions.

To address model underfitting in Bayesian nonparametric regression arising from a finite candidate model space, this paper proposes a novel prior distribution that integrates roughness penalty and ridge penalty—marking the first incorporation of penalized splines into the Bayesian model selection framework. This convex-combination penalty prior adaptively controls both function smoothness and the number of basis functions without requiring prespecified smoothness parameters. Theoretically, the posterior contraction rate achieves the minimax optimal rate (up to a logarithmic factor). Computationally, efficient inference is enabled via Markov Chain Monte Carlo. Simulation studies and real-data applications demonstrate that the method achieves superior balance between predictive accuracy and model parsimony, significantly outperforming existing Bayesian spline approaches while providing rigorous theoretical convergence guarantees.

This work addresses the computational challenges in existing Bayesian factor models arising from the complex hierarchical structure of ordered shrinkage priors, which hinder efficient posterior inference. To overcome this limitation, we propose a novel Bayesian factorization method based on an $L_{1/2}$ shrinkage prior that preserves the desirable ordered shrinkage property of factor loadings while substantially simplifying the prior architecture. The resulting model admits both exact Gibbs sampling and an efficient variational approximation, achieving a favorable balance between computational efficiency and inferential accuracy. Extensive numerical experiments demonstrate that the proposed approach consistently outperforms state-of-the-art Bayesian factor models in terms of both estimation precision and computational speed.

This paper investigates the “information borrowing” mechanism driven by shared hyperparameters in hierarchical Bayesian models and its impact on posterior inference performance. Method: Focusing on mixed-effects models, we propose an integrated risk measure grounded in the true data-generating distribution and develop a non-asymptotic theoretical framework to quantify information borrowing efficiency under varying prior depths. Contribution/Results: We prove that when random effects exhibit a compound-symmetric structure—especially with strong between-group correlation—the Bayesian posterior mean estimator from a deeply nested hierarchical model strictly dominates that from any shallower nested submodel; we further derive necessary and sufficient conditions for this dominance. The result extends to perturbed correlation structures. To our knowledge, this is the first work to systematically characterize the statistical gains from information borrowing in a non-asymptotic setting, providing both theoretical justification and practical criteria for selecting optimal prior depth in hierarchical modeling.

This study investigates the impact of default prior choices on risk in high-dimensional shrinkage estimation, with a focus on the differing behavior of priors near zero under variance versus standard deviation parameterizations. Through high-dimensional asymptotic analysis, radial power benchmarks, and characterization via the zero-index of scale densities, the work provides the first geometric insight showing that a flat prior on the standard deviation enjoys a unit asymptotic risk advantage near the origin, yielding lower risk under weak signals and second-order equivalence to the flat variance prior under strong signals. The paper further establishes precise risk crossover and equivalence relationships between these two prior classes and offers a unified classification framework for heavy-tailed or sparse priors.

This work addresses the challenge of designing priors for Bayesian neural networks that simultaneously promote sparsity, robustness, and predictive performance while mitigating the cold posterior effect. The authors propose a Dirichlet Scale Mixture (DSM) prior that leverages its heavy-tailed nature and structured sparsity-inducing shrinkage mechanism to enable implicit feature selection and parameter reduction. This approach effectively alleviates the cold posterior problem and enhances model robustness and prunability, particularly under small-sample regimes and with correlated data. Empirical evaluations on both synthetic and real-world datasets demonstrate that the DSM prior achieves competitive predictive accuracy while substantially reducing the effective number of parameters and improving robustness against adversarial attacks, offering a superior paradigm for sparse prior specification in Bayesian neural networks.

This work addresses the high computational cost of Bayesian inference with projection priors, which typically require nested iterative optimization. The authors propose a continuously relaxed projection prior that eliminates the need for inner-loop optimization during posterior updates by introducing a duality gap and placing a probabilistic prior on it to drive its contraction toward zero. This approach is the first to incorporate a continuous shrinkage mechanism into projection priors, yielding a differentiable, inner-loop-free approximate prior. The method also establishes a theoretical connection to global–local shrinkage priors. Empirical results demonstrate competitive posterior shrinkage performance, and the approach is successfully applied to marketing data analysis of multivariate shopping decisions, effectively identifying key predictive factors.

Traditional Bayesian modeling relies on model selection to balance complexity and generalization, yet this approach often compromises predictive performance in small-sample settings. This work proposes a “predictive consistency prior” that maintains stability in the prior predictive distribution as model complexity increases, thereby circumventing explicit model selection. By shifting the modeling focus from parameter sparsity to constructing reasonable and stable priors in predictive space, the method reveals that the perceived necessity of model selection fundamentally stems from inadequate prior specification. The authors implement this prior in Bayesian linear and logistic regression, forward variable selection, and nonlinear models, demonstrating through numerical experiments that flexible models equipped with the predictive consistency prior match or even outperform carefully selected simpler models in out-of-sample prediction across a range of tasks.

This work addresses the challenges in high-dimensional Bayesian regression, where conventional MCMC methods often get trapped in local modes and maximum a posteriori (MAP) estimation fails to quantify uncertainty. To overcome these limitations, the authors propose a hybrid approach that integrates deterministic optimization with stochastic sampling. Specifically, under a heavy-tailed hyperbolic error model, they first employ a two-stage ECM algorithm to efficiently perform variable selection and substantially reduce the model space. Subsequently, Gibbs sampling is conducted within the high posterior probability subspace to enable full posterior inference, complemented by Bayesian model averaging. The proposed method effectively balances computational efficiency, variable selection accuracy, and robust uncertainty quantification. Empirical evaluations on both simulated and real-world datasets demonstrate its superior performance over current state-of-the-art methods.