posterior predictive tilting

This study addresses the failure of classical Monte Carlo algorithms in prediction-oriented posteriors due to the absence of an explicit density. We propose a pointwise density approximation method that integrates Markov chain Monte Carlo with numerical approximation techniques, enabling posterior sampling with rapid error decay. This approach overcomes the sampling bottleneck for distributions lacking closed-form densities and effectively quantifies uncertainty even under model misspecification. The proposed method is validated across diverse applications, including epidemiology, spatial statistics, and nuclear physics, demonstrating both effectiveness and high precision. Ultimately, this work establishes a robust new paradigm for complex Bayesian inference.

This study addresses the lack of theoretical guarantees regarding the impact of predictive design distributions on inference validity in high-dimensional Bayesian regression. The authors propose a novel parametric martingale posterior approach based on sequential one-step-ahead predictive quantities, which operates without Markov chain Monte Carlo methods and, for the first time, systematically elucidates the critical role of predictive design distributions. The method satisfies weak identifiability and design invariance, and integrates predictive resampling with high-dimensional regularization techniques. In simulation experiments, it demonstrates both computational efficiency and stable performance in Bayesian predictive inference.

This work addresses predictive calibration under label shift for Tobit Gaussian regression with doubly censored responses. The authors propose a conformal Bayesian framework tailored to mixed response spaces that combine point masses at boundary atoms and continuous densities over interior intervals. By integrating latent-variable exponential tilting with weighted conformal calibration, the method leverages Laplace approximations to the posterior to derive a calibration mechanism that jointly accounts for boundary weights and interior density ratios. This yields adaptive, geometrically structured highest-density prediction sets for mixed outcomes. Experiments on synthetic data demonstrate that the approach restores marginal coverage while producing substantially smaller prediction sets than conventional weighted source-score calibration, and further uncover an intrinsic trade-off between coverage performance and the relative contributions of atomic versus continuous components in the response distribution.

This study addresses the challenge of accurate parameter estimation in Bayesian inference when selection bias and systematic bias are present. The authors propose a generalized Bayesian approach that, for the first time, integrates inverse probability weighting (IPW) into the Bayesian framework. By interpreting IPW as a reweighting of the Kullback–Leibler divergence between the model and the true data-generating mechanism, they construct a posterior distribution that simultaneously inherits the bias-correction properties of frequentist IPW and maintains Bayesian coherence. Theoretical analysis establishes favorable asymptotic convergence properties of the proposed posterior. Empirical validation on both simulated data and a large-scale prostate cancer registry dataset—used to predict mortality based on PSA levels—demonstrates the method’s effectiveness, substantially extending the applicability of Bayesian inference to settings with biased observations.

This study addresses the failure of conformal prediction coverage guarantees caused by post-deployment data distribution shifts. It proposes Exponential Tilting Reweighting Alignment (ExTRA), a framework for modeling distribution shifts that systematically compares two adaptation strategies: weight calibration and prediction tilting. The analysis reveals that while prediction tilting can reduce set sizes under specific conditions, it may compromise coverage validity. Experiments demonstrate that ExTRA decreases prediction set lengths by 30% in synthetic regression tasks; however, it induces significant coverage degradation in classification or information-deficient settings and yields no consistent benefits on real-world data. Determining the precise applicability conditions for this approach remains an open challenge.

This work proposes a prediction-oriented Bayesian inference approach under model misspecification, which constructs a posterior distribution that balances predictive performance and uncertainty quantification by optimizing a scoring rule—such as the logarithmic score—over predictive distributions, augmented with a φ-divergence regularizer relative to a reference prior. Leveraging a finite-dimensional dual formulation, the method establishes theoretical optimality under a zero duality gap condition and derives finite-sample bounds on predictive risk for the resulting approximate posterior. By employing a semi-analytical posterior representation and solving the associated dual optimization problem, the approach demonstrates superior predictive accuracy and numerical stability in both classification tasks and experiments involving Gaussian mixture misspecification.

This study addresses the limitation of traditional quantile regression, which requires prespecifying endpoint quantiles and thus struggles to flexibly characterize optimal intervals with fixed probability content but variable location—such as equal-tailed or shortest continuous intervals. The authors propose a mean-tilting framework that relaxes the check loss of quantile regression to estimate unlabeled interval root functionals and explicitly links interval location to internal mean shifts. Building on this, they develop a loss-based generalized Bayesian inference approach, integrating pseudo-asymmetric Laplace normal-exponential augmentation, Nishimura–Suchard augmentation, and Gaussian prior samplers, thereby accommodating both static regression and dynamic state-space extensions. The standard relaxed quantile regression algorithm has been implemented and validated for root functional estimation; the non-zero tilting algorithm has been derived and awaits implementation and empirical verification.

This study addresses the limitation of existing recalibration methods, which often obscure miscalibration in specific regions such as extreme events. To overcome this, we propose an outcome-conditional recalibration post-processing method that leverages quantile recalibration and conditional distribution scaling to achieve precise correction of arbitrary predictive distributions within user-defined regions. By simultaneously preserving global performance and local reliability, the proposed approach significantly enhances conditional calibration on regression benchmark tasks. Furthermore, when applied to electricity price forecasting, it substantially improves calibration in negative-price regimes with negligible accuracy loss. Overall, this work provides more reliable localized guarantees for probabilistic forecasting.

This study addresses the quantification of posterior uncertainty in kernel density estimation within a predictive Bayesian framework. By analyzing the predictive measure induced by kernel density estimators, the authors establish, for the first time, that the associated resampling sequence—despite failing to satisfy conditional independence and identical distribution (c.i.d.) or asymptotic c.i.d. (a.c.i.d.) conditions—converges weakly almost surely. In the case of Gaussian kernels, they further derive an explicit density representation of the limiting random probability measure and construct corresponding moment estimators. Leveraging these results, the paper successfully derives Bayesian credible intervals for kernel density estimates and demonstrates their empirical validity on two real-world datasets, thereby providing a rigorous tool for uncertainty quantification in nonparametric density estimation.

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.