posterior convergence analysis

This study addresses Bayesian inference for low-dimensional target parameters in semiparametric models, particularly under the presence of complex nuisance components that may compromise frequentist properties. To this end, we construct posterior distributions by integrating estimating function methods with nonparametric Bayesian techniques—such as Dirichlet processes and Bayesian bootstrap—under conditions weaker than the classical stochastic equicontinuity assumption. We establish asymptotic normality and consistency of the resulting posterior, rigorously identifying the key assumptions required to guarantee desirable frequentist behavior. The theoretical analysis systematically elucidates how relaxing these assumptions affects inferential performance. Extensive simulations corroborate the effectiveness of the proposed methodology, demonstrating its robustness and accuracy in practical settings.

This work investigates the non-asymptotic characterization of posterior contraction rates and finite-sample Bernstein–von Mises (BvM) behavior in nonparametric Bayesian models. By interpreting the posterior distribution as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, the study extends diffusion-based analytical frameworks to infinite-dimensional settings for the first time. This approach enables precise control over posterior moments and yields sharp non-asymptotic concentration rates in Hilbert norm, along with a quantitative Laplace approximation of the posterior. In the context of nonparametric linear Gaussian inverse problems, the theory elucidates how likelihood curvature and prior regularity jointly govern posterior contraction rates and finite-sample BvM phenomena.

This paper addresses the lack of frequentist guarantees and robustness theory for generalized posteriors (M-posteriors). Methodologically, it introduces novel definitions of the posterior influence function and posterior breakdown point, and conducts asymptotic analysis integrating M-estimation loss functions with prior conditions. Under mild regularity assumptions, it establishes that M-posteriors are asymptotically normal, concentrate around the corresponding M-estimator, and simultaneously achieve frequentist consistency and Bayesian interpretability. The key contributions are: (i) the first systematic robustness characterization of M-posteriors, formalized via influence functions and breakdown points; (ii) a natural extension of classical M-estimation robustness to Bayesian posterior inference; and (iii) theoretical validity across broad settings—including generalized linear models and quantile regression—supported by numerical experiments demonstrating stability under outliers and model misspecification.

Stochastic MALA (sMALA) suffers from posterior drift—deviating from the true Gibbs posterior toward a non-Gibbs target—due to minibatch-induced bias, thereby compromising uncertainty quantification in Bayesian neural networks. Method: We propose a stochastic Metropolis–Hastings algorithm with an explicit, computable correction term that rigorously restores sampling from the original Gibbs posterior. This is the first MH-based stochastic sampler to incorporate an analytically tractable bias correction. Contributions/Results: We establish a PAC-Bayesian theoretical foundation for deep nonparametric regression, proving that the corrected algorithm achieves optimal posterior contraction rates and yields credible sets with guaranteed high coverage probability. Numerical experiments demonstrate that its uncertainty quantification performance matches classical MALA and significantly outperforms uncorrected sMALA, validating both statistical fidelity and practical efficacy.

This work addresses posterior functional computation in high-dimensional generalized linear models under non-log-concave likelihoods. To overcome the lack of non-asymptotic theoretical guarantees for conventional MCMC methods under non-convex posteriors, we propose a gradient-driven MCMC algorithm that achieves statistically optimal posterior sampling in polynomial time, requiring only local likelihood regularity and a suitable initial point. Our contribution is the first non-asymptotic convergence theory for posterior sampling that operates outside the M-estimation framework—free from asymptotic assumptions and scalable to high dimensions—applicable to density estimation, nonparametric regression, and PDE inverse problems. Crucially, the theoretical guarantees hold rigorously under non-log-concave likelihoods. Empirical evaluations confirm the method’s effectiveness and computational efficiency in both generalized linear models and PDE inverse problems.

This study addresses the vulnerability of traditional Bayesian posterior inference to model misspecification, which arises when the likelihood function fails to accurately characterize the underlying data-generating mechanism. To overcome this limitation, this work reconstructs Bayesian theory from a variational perspective and proposes a generalized Bayesian inference framework. By integrating variational inference with robust statistical techniques, it establishes a novel paradigm for variational posteriors under model misspecification. This research effectively mitigates the challenges posed by misspecified models, substantially broadening the applicability of Bayesian inference. Furthermore, it enhances both the reliability and precision of uncertainty quantification and parameter estimation in non-ideal modeling conditions.

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 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.

本文探讨了贝叶斯样本量确定的两种方法：估计抽样分布或通过随机根查找探索，评估了它们在复杂模型中的性能。

This study addresses the asymptotic behavior of Bayesian posteriors in high-dimensional generalized linear models when the number of features grows proportionally with the sample size. Despite concerns about posterior contraction failure and unclear finite-dimensional marginal behavior in this regime, the authors establish—via leave-one-out analysis, high-dimensional asymptotic theory, and explicit posterior derivations—that the marginal posterior converges to a prior-induced Gaussian tilt distribution, whose mean depends on the true signal coordinates. This result demonstrates the persistent influence of the prior in high-dimensional Bayesian inference and further shows that the posterior mean strictly dominates the maximum likelihood estimator in terms of mean squared error, uniformly across all sparsity levels.