This survey article reviews Bayesian quadrature, a probabilistic, model‑based method for numerical integration and expectation estimation. It provides a systematic taxonomy of Bayesian quadrature techniques across modelling, inference, and sampling, presents theoretical guarantees, and includes a controlled numerical study illustrating the impact of different methodological choices. The paper also discusses practical challenges, limitations, and offers an extensive bibliography covering machine learning, statistics, mathematics, and engineering applications.
arXiv:2607. 10793v1 Announce Type: new Abstract: Numerical integration is a cornerstone of various scientific computing applications, such as engineering simulations and model evidence computations in probabilistic machine learning.
arXiv:2606.15871v2 Announce Type: replace-cross Abstract: Uncertainty in the solution of an inverse problem and in the tasks performed on it is quantified by posterior expectations, each an average o...
The paper introduces a two-step Metropolis–Hastings algorithm designed to efficiently sample from Bayesian empirical likelihood (BayesEL) posterior distributions, addressing challenges posed by the complex, often non‑convex support of empirical likelihood. The method leverages current parameter values and estimating equations to propose new values for remaining parameters, making it suitable for problems with discontinuous estimating equations such as simultaneous quantile regression. Additionally, the approach extends naturally to BayesEL model selection via reversible‑jump MCMC, and the authors demonstrate its utility through several real‑life applications.
arXiv:2602. 19126v2 Announce Type: replace Abstract: We propose a robust Bayesian formulation of random feature (RF) regression that accounts explicitly for prior and likelihood misspecification via Huber-style contamination sets.
The paper introduces a Bayesian approach to matrix completion that uses a nuclear norm-based prior and addresses the challenge of unknown noise variance by placing a prior on it. It presents the first sampler for this model, providing a non‑asymptotic polynomial‑time guarantee in terms of matrix dimensions and desired accuracy. The method discretizes the noise precision and employs thermodynamic integration to construct a categorical posterior, offering a feasibility result for Bayesian sampling in non‑log‑concave settings.
arXiv:2606. 27269v1 Announce Type: cross Abstract: Reliably quantifying predictive uncertainty is difficult for complex, high-dimensional, or misspecified models.
The paper introduces SME-BETEL, a semiparametric Bayesian method that merges score matching estimating equations with Bayesian exponentially tilted empirical likelihood to perform inference on models with intractable normalizing constants. SME-BETEL avoids evaluating these constants and eliminates the need for learning-rate calibration, while providing consistency, asymptotic normality, and a Bernstein‑von Mises theorem that guarantees asymptotically calibrated credible sets even under model misspecification. The authors extend the framework to mixed‑domain data, enabling robust inference for doubly‑intractable models such as spatial preferential sampling, and demonstrate its effectiveness through simulations and an ozone‑monitoring application.
arXiv:2402. 11736v3 Announce Type: replace Abstract: Kernel herding belongs to a family of deterministic quadratures that seek to minimize the maximum mean discrepancy (MMD), that is, the worst-case integration error over a reproducing kernel Hilbert space (RKHS).
arXiv:2504. 01894v2 Announce Type: replace Abstract: We present a bifidelity method for uncertainty quantification of parameter estimates in complex systems, leveraging generative models trained to sample the target conditional distribution.
arXiv:2608.23802v1 Announce Type: cross Abstract: Many common data dependencies can be characterized by graphs: time series data are sequential (chain graph), images appear as pixels (lattice graph),...
arXiv:2609.39712v1 Announce Type: cross Abstract: We consider simulation-based Bayesian inference (SBI) for the parameters of models with intractable likelihoods but tractable forward simulation. Bui...
arXiv:2607. 19498v1 Announce Type: cross Abstract: Gaussian process (GP) modeling is widely used in computational science and engineering.