analytical marginalization

Unbinned likelihood fits aim at maximizing the information one can extract from experimental data, yet their application in realistic statistical analyses is often hindered by the computational cost of profiling systematic uncertainties. Additionally, current machine learning-based inference methods are typically limited to estimating scalar parameters in a multidimensional space rather than full differential distributions. We propose a general framework for Simulation-Based Inference (SBI) that efficiently profiles nuisance parameters while measuring multivariate Distributions of Interest (DoI), defined as learnable invertible transformations of the feature space. We introduce Factorizable Normalizing Flows to model systematic variations as parametric deformations of a nominal density, preserving tractability without combinatorial explosion. Crucially, we develop an amortized training strategy that learns the conditional dependence of the DoI on nuisance parameters in a single optimization process, bypassing the need for repetitive training during the likelihood scan. This allows for the simultaneous extraction of the underlying distribution and the robust profiling of nuisances. The method is validated on a synthetic dataset emulating a high-energy physics measurement with multiple systematic sources, demonstrating its potential for unbinned, functional measurements in complex analyses.

This work addresses the challenges of high-dimensional Bayesian inverse problems, which often arise due to unknown physical mechanisms, difficulty in quantifying measurement uncertainties, or prohibitive costs of high-fidelity simulations that render conventional inference methods ineffective. The authors propose a neural likelihood approximation framework that directly learns an unnormalized likelihood from data while explicitly incorporating the normalization constant into the training objective. They establish, for the first time, that this learning problem constitutes a strictly convex optimization task. By leveraging KL divergence minimization and empirical risk minimization, they develop a consistency theory guaranteeing uniform convergence of the neural likelihood estimator. Empirical results on image deblurring and nonlinear PDE-based imaging demonstrate that the method stably converges to the true likelihood as sample size increases, significantly enhancing the scalability and robustness of Bayesian inversion.

This work addresses the ill-posed inverse problems arising from highly nonlinear forward models, the coexistence of additive and multiplicative noise, and censored observations. To tackle these challenges without approximating the likelihood function or neglecting any noise source, the authors propose a general hierarchical Bayesian modeling framework. Coupled with an efficient Markov chain Monte Carlo (MCMC) algorithm, the method directly samples from the complex posterior distribution, thereby avoiding biases introduced by model simplifications or ad hoc hyperparameter tuning in conventional approaches. Extensive experiments on synthetic astronomical data demonstrate that the proposed method consistently outperforms existing baselines and state-of-the-art techniques in terms of point estimation accuracy, predictive performance, and computational efficiency, achieving state-of-the-art uncertainty quantification and reliable inference.

In Bayesian inverse problems, unreliable uncertainty quantification arises from manually specified priors and inaccurate forward models. To address this, we propose a novel method for automatically learning parameter prior distributions directly from noisy indirect observational data. Our approach introduces a bilevel optimization framework: the upper level learns a generative prior mapping—specifically, a Gaussian pushforward in latent space—while the lower level jointly trains a residual neural operator as the forward model. We design a computationally tractable loss function based on empirical approximation of a divergence metric. Evaluated on Darcy flow permeability inversion, the method significantly reduces reliance on ad hoc smoothness assumptions about the prior and on exact knowledge of the forward physics. It enables efficient and robust prior calibration, thereby enhancing both the reliability and generalizability of posterior uncertainty quantification.

To address insufficient Bayesian prior information in few-shot personalized inference, this work proposes a novel paradigm for constructing transferable, informative priors from population data. The core method introduces data-consistent stochastic inversion (DCI) to learn pullback probability measures from population-level observations, thereby deriving structured Gaussian priors tailored to individual inverse problems. Theoretically, we prove that the resulting prior strictly improves information gain—measured by both determinant and trace of the posterior precision matrix—and reduces the posterior KL divergence in linear Gaussian inverse problems. Numerical experiments demonstrate substantial improvements in inference accuracy and uncertainty calibration across digital twin and biomedical modeling tasks. This work establishes a provably sound, transferable, and physics-agnostic framework for data-driven prior construction.

This work addresses the limitations in multiparameter Bayesian inversion arising from oversimplified assumptions of parameter independence or regularization strategies lacking statistical justification. It proposes a joint Gaussian prior construction that preserves prescribed marginal Gaussian distributions while incorporating spatially varying cross-correlation structures via a principal square-root covariance decomposition. This formulation is optimal in the sense of canonical correlation analysis and enables explicit quantification of uncertainty in the correlation structure itself. By combining a rigorously contractive mapping with Bayesian sampling, the method facilitates efficient posterior inference. Numerical experiments demonstrate that neglecting either parameter correlations or their uncertainties leads to substantial estimation bias, thereby validating the necessity and efficacy of treating unknown parameters as random variables with explicitly modeled dependence structures.

This study addresses the lack of uncertainty quantification and statistical inference methods for high-dimensional tensor completion. Focusing on low tubal-rank tensor completion, the work proposes the first framework enabling rigorous statistical inference by constructing asymptotically Gaussian estimators through two-sample debiasing, low-rank projection, and frequency-domain low tubal-rank modeling. This approach facilitates element-wise confidence intervals and hypothesis testing. Theoretical analysis establishes the asymptotic normality and statistical validity of the proposed estimator. Experiments on both synthetic data and real-world global ionospheric total electron content measurements demonstrate that the resulting confidence intervals are robust and reliably capture the intrinsic variability of the data.

This work proposes a novel Bayesian framework that integrates Bayesian optimization with Bayesian inversion to address inverse problems in scenarios where high-fidelity models are computationally expensive and observational data are limited. By adaptively constructing a Gaussian process surrogate model, the method jointly leverages prior information and observed data to achieve efficient parameter inference and robust uncertainty quantification. The key innovation lies in the synergistic coupling of Bayesian optimization and Bayesian inversion, which significantly enhances both the efficiency of surrogate model construction and the accuracy of the inversion. Experimental results on multiple benchmark test functions demonstrate that the proposed framework achieves high-precision parameter estimates and reliable uncertainty quantification while substantially reducing computational cost, thereby offering an efficient and trustworthy tool for engineering decision-making.

This work addresses the ill-posedness of inverse problems governed by partial differential equations, which arises from data noise, missing observations, and non-uniqueness, and for which existing Bayesian methods struggle to enforce hard physical constraints effectively. The authors propose a dual-space sampling framework that uniquely integrates the augmented Lagrangian method, the alternating direction method of multipliers (ADMM), and Stein variational gradient descent (SVGD) to transform hard constraints into differentiable penalty terms. This approach enables efficient posterior sampling while strictly satisfying physical laws. It combines the well-conditioned nature of dual solvers with the nonparametric expressiveness of SVGD. Experiments on Rosenbrock inference, Gaussian anomaly modeling, and Marmousi II full-waveform inversion demonstrate that the method yields well-calibrated uncertainty estimates, with posterior distributions converging stably as data coverage increases.

This work addresses inverse problems in science and engineering—such as parameter inference and detector response unfolding—by proposing a unified simulation-based inference (SBI) framework that systematically integrates Bayesian and frequentist perspectives. Leveraging machine learning techniques, including neural posterior estimation and neural likelihood estimation, the framework enables efficient and general-purpose parameter inference, with extensions to empirical Bayes and unfolding tasks. The paper provides a comprehensive review of SBI methodologies and their application paradigms, while also offering a thorough analysis of validation strategies and inherent limitations. By clarifying best practices and pitfalls, this study advances the reliable deployment and innovative application of SBI in scientific domains.