posterior credible intervals

Approximate Bayesian inference often underestimates true uncertainty due to posterior credible intervals that are excessively narrow. This work proposes two simulation-based calibration (SBC)-driven methods for recalibrating approximate posteriors, systematically leveraging the SBC framework to adjust the width of posterior uncertainty intervals and achieve marginal calibration. The approach is applicable to complex model structures, including hierarchical models, and demonstrates consistent efficacy across diverse experimental settings by meaningfully widening posterior intervals. As a result, the proposed recalibration substantially enhances the calibration accuracy and reliability of approximate Bayesian inference.

This study addresses the challenge of constructing statistical intervals that simultaneously admit a Bayesian interpretation and satisfy frequentist finite-sample coverage guarantees, all within a fully nonparametric setting. To bridge the gap between Bayesian and frequentist paradigms, the work proposes a relaxed notion of Bayesian credible intervals: rather than requiring a fixed posterior probability ex ante, it only demands that, after observing the interval, the posterior confidence be at least $p\%$. Methodologically, the approach introduces a one-dimensional prior to circumvent the complexities of high-dimensional modeling and leverages a decision-theoretic framework combined with nonparametric estimation techniques to construct intervals for both cumulative distribution functions and means of distributions with bounded support. The resulting intervals are asymptotically equivalent to those from full Bayesian procedures or slightly wider, yet they offer stronger finite-sample coverage assurances.

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.

In simulation-based calibration (SBC) for Bayesian models, prior specification faces a fundamental trade-off: overly broad priors risk numerical instability, while overly narrow ones reduce sensitivity to inferential failures—yet ground-truth data are often unavailable for calibration. Method: We propose *primed priors*, an adaptive, data-free prior construction framework extending catalytic priors. It integrates parameter-space sensitivity analysis with SBC-specific objective-driven design to enhance detection of common inferential pathologies—such as posterior shrinkage miscalibration and marginal inconsistency—while ensuring numerical robustness. Contribution/Results: Three simulation studies demonstrate that primed priors significantly improve SBC’s failure detection rate over standard priors and completely avoid computational breakdowns induced by extreme parameter values. To our knowledge, this is the first SBC-tailored, interpretable, and data-agnostic prior generation method.

This work addresses the challenge of conducting calibrated Bayesian inference for parametric models whose likelihood functions are intractable, numerically unstable, or computationally prohibitive. Existing approaches lack finite-sample calibration guarantees under such conditions. The authors propose a fully probabilistic inference framework that requires neither a prior nor a likelihood, relying solely on the model’s simulation capability. By leveraging permutation-invariant functions—such as depth functions—to rank parameters and introducing a closed-form rescaling procedure, the method achieves finite-sample frequentist calibration. To the best of the authors’ knowledge, this is the first approach to provide theoretical calibration guarantees in a setting devoid of both likelihood and prior specifications. Empirical evaluations across four benchmark tasks—including differential privacy and the Ising model—as well as a spatial analysis of the 2025 U.S. measles outbreak demonstrate the method’s strong practical utility and robustness.

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 study addresses the challenge of inaccurate estimation in recursive Bayesian inference when significant inter-stage shifts occur in the posterior distribution. To overcome this limitation, the authors propose and implement a Parallel Annealed Prior-Proposal Recursive Bayesian (PPP-RB) method that integrates recursive Bayesian updating, parallel computation, and an annealing mechanism inspired by Metropolis-coupled Markov chain Monte Carlo. PPP-RB maintains accurate approximation of the true posterior across multiple updating stages, effectively resolving the failure of existing prior-proposal recursive Bayesian (PP-RB) approaches under substantial posterior shifts. The method provides theoretical guarantees for correctness of the target posterior while substantially improving computational efficiency. Empirical evaluations on earthquake count data and North Atlantic sea surface salinity measurements demonstrate that PPP-RB yields a higher effective sample size per unit time compared to both PP-RB and standard MCMC algorithms.

This study addresses the long-standing challenge of constructing nontrivial, efficient confidence intervals for the location parameter of a location-scale family when only a single observation is available. The authors propose two Bayesian approaches: first, deriving priors that yield asymptotically efficient intervals at high confidence levels; second, integrating classical t-intervals with prior information to form an enhanced t-interval based on Bayes factor testing. The work establishes, for the first time, a systematic Bayesian mechanism for generating single-sample confidence intervals and demonstrates an equivalence between Bayesian credible intervals and frequentist confidence intervals. The methodology extends to any continuous symmetric location-scale family. Theoretical results show that the proposed intervals are asymptotically efficient when \( n = 1 \), and for \( n \geq 2 \), the enhanced t-interval achieves smaller expected squared width over parts of the parameter space, with practical utility validated on interstellar object velocity data.

This work addresses the non-uniform distribution of posterior predictive p-values (ppp) under the Bayesian framework, which hinders reliable model diagnostics and cross-model comparisons. The authors propose a natural calibration method that transforms ppp values into calibrated posterior predictive p-values (cppp), which follow a standard uniform distribution under the true model. This calibration establishes, for the first time, a unified and comparable scale for ppp-based assessments. The approach is grounded in a double-simulation computational framework that seamlessly integrates Bayesian inference with posterior predictive checking, and it is applicable to both parametric and nonparametric models. Theoretical analysis demonstrates favorable statistical properties of cppp, while empirical studies illustrate its effectiveness in enabling fair comparisons among models and prior specifications on real-world data.