statistics

This study addresses the challenge of conducting valid Bayesian inference for target parameters—such as causal effects—in the presence of finite-dimensional nuisance parameters. The authors propose a general framework that integrates Bayesian bootstrap with Dirichlet process priors within an estimating equation approach, explicitly accounting for uncertainty in propensity score estimation. The method is robust to model misspecification and remains valid under only single robustness conditions. It extends the "linked Bayesian bootstrap" to nonstandard Bayesian settings, yielding posterior inferences with favorable frequentist properties. Theoretical analysis demonstrates that the resulting posterior distribution exhibits desirable asymptotic behavior, with credible intervals achieving nominal coverage probabilities in large samples.

To address the lack of theoretical foundation for “uncertainty in uncertainties” in experimental data fusion, this paper establishes a unified probabilistic framework that rigorously models uncertainty in quoted variances, thereby bridging the conceptual gap between Bayesian and frequentist approaches. By introducing auxiliary gamma-distributed variables, we construct a coherent statistical model and formally prove one-to-one parameter correspondence and structural equivalence between the two paradigms. This constitutes the first rigorous characterization of the intrinsic relationship between Bayesian prior specification and frequentist sampling assumptions. The framework provides a solid theoretical basis for applications in particle physics and related fields, and enables methodological interchangeability—allowing practitioners to seamlessly translate between Bayesian and frequentist implementations. As a result, it significantly enhances both the reliability and interpretability of combined analyses involving inconsistent measurements.

Traditional hybrid experimental designs struggle to robustly control the frequentist operating characteristics of Bayesian decisions under model misspecification and lack efficient sample size determination methods applicable to generalized posteriors. This work proposes a computationally efficient experimental design framework that requires simulations at only two sample sizes and leverages extrapolation modeling of posterior summary functions to infer performance across the entire sample size space. This approach enables identification of the minimal sample size and decision rule satisfying desired operating characteristics. It represents the first general and scalable method for sample size planning under generalized posteriors, substantially reducing computational burden while enhancing robustness to model misspecification. The method’s validity and broad applicability within Bayesian M-estimation–type experiments are demonstrated through the redesign of an adaptive clinical trial with time-to-event outcomes.

This paper addresses the complexity and high pedagogical/practical barriers associated with conventional uncertainty quantification methods—such as standard errors, confidence intervals, and hypothesis tests—in statistical inference. To evaluate the potential of nonparametric bootstrap as a unified alternative, we conduct a large-scale simulation study rigorously comparing single bootstrap, double bootstrap, and classical methods across multiple dimensions: sample size, confidence level, data-generating mechanisms, and statistical functionals. Results demonstrate that the double bootstrap consistently achieves superior coverage accuracy, stability, and robustness—particularly under small-sample and non-normal conditions—outperforming both classical approaches and the single bootstrap. We thus establish the double bootstrap as a principled, parsimonious, and high-performance paradigm for uncertainty quantification, providing both theoretical justification and empirical evidence to support its adoption in statistical education and applied practice.

This work addresses the long-standing lack of a systematic theoretical foundation for confidence distributions (CDs) within the frequentist framework, which has hindered their broader adoption. By integrating and extending the theoretical underpinnings of CDs—drawing on Fisher’s concept of fiducial likelihood and modern frequentist inference techniques—the project establishes a unified framework for parametric inference. Through the systematic development of its methodology and promotion of interdisciplinary applications, the study not only strengthens the theoretical basis of CDs but also revitalizes their role in statistical practice. The findings synthesize cutting-edge research presented at the 2015 “Inference With Confidence” workshop in Oslo, significantly enhancing the influence and practical utility of confidence distributions in the statistical community.

This study addresses the challenge of effectively introducing Bayesian inference in introductory statistics courses while minimizing computational complexity and fostering active learning. To this end, the authors developed a classroom-ready instructional activity grounded in Process-Oriented Guided Inquiry Learning (POGIL), which employs two-way probability tables to guide students through manual calculations of conditional probabilities and belief updating, implemented via structured small-group roles. A reproducible Bayesian assessment framework was also created, with learning outcomes analyzed using bivariate generalized linear models. Empirical results demonstrate that this approach yields exam performance and instructional satisfaction comparable to traditional lecture-based methods, with consistent efficacy across majors, genders, and racial groups, thereby affirming its feasibility, inclusivity, and potential for broad adoption.

This work addresses the challenge of uncertainty quantification in Poisson signal models with background noise by proposing a confidence interval construction grounded in the principle of Bayesian evidence and the framework of relative belief inference. The method achieves both Bayesian interpretability and frequentist coverage guarantees without requiring prior information, while preserving likelihood ordering consistency and rigorously attaining the prescribed coverage probability. In benchmark scenarios commonly encountered in particle physics, the proposed intervals outperform the widely used Feldman–Cousins approach, thereby offering superior statistical performance. Notably, this is the first method to successfully unify a Bayesian evidential interpretation with strict frequentist coverage properties.

This work addresses the lack of explicit confidence modeling for various sources of uncertainty in Bayesian inference by proposing a general extension framework that, for the first time, explicitly incorporates confidence in key uncertainty components—such as the prior and likelihood—into Bayesian modeling. The framework not only introduces a novel regularization mechanism but also provides a unified approach to inducing model sparsity. Without compromising theoretical rigor, the method achieves controllable sparsity across diverse models, including linear regression, logistic regression, and Bayesian neural networks, thereby significantly enhancing both interpretability and generalization performance.

This work addresses the limitations of traditional Bayesian methods in model flexibility and adaptability to complex data by systematically integrating the core theory and recent advances in Bayesian nonparametrics. It establishes a unified theoretical and practical framework, with particular emphasis on key models such as the Dirichlet process and Gaussian process, along with their associated probabilistic modeling and inference techniques. The primary outcome is an authoritative monograph that comprehensively articulates the theoretical foundations, representative applications, and future research directions of the field. By doing so, it significantly enhances the accessibility and depth of application of Bayesian nonparametric methods, providing a robust reference for researchers and practitioners alike.