SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

• URL: http://arxiv.org/abs/2603.22468v1
• Date: Mon, 23 Mar 2026 18:36:36 GMT
• Title: SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation
• Abstract summary: We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models.<n>We also establish a quantitative Laplace approximation for the posterior.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: We derive posterior contraction rates (PCRs) and finite-sample Bernstein von Mises (BvM) results for non-parametric Bayesian models by extending the diffusion-based framework of Mou et al. (2024) to the infinite-dimensional setting. The posterior is represented as the invariant measure of a Langevin stochastic partial differential equation (SPDE) on a separable Hilbert space, which allows us to control posterior moments and obtain non-asymptotic concentration rates in Hilbert norms under various likelihood curvature and regularity conditions. We also establish a quantitative Laplace approximation for the posterior. The theory is illustrated in a nonparametric linear Gaussian inverse problem.

Related papers

• Posterior contraction rates in Sobolev norms and Bayesian derivative estimation for infinite-dimensional exponential families [6.531568167097593]
  We study posterior contraction in positive-order Sobolev norms and Bayesian derivative estimation for exponential families.<n>We show that smoothness-matching priors achieve minimax-optimal posterior contraction rates in any Hilbert scale norm up to the regularity of the ground truth.
  arXiv  Detail & Related papers  (2026-08-11T16:45:55Z)
• Spectral Bayesian Regression on the Sphere [0.0]
  We develop a framework for nonparametric regression on the unit sphere based on isotropic Gaussian field priors and the harmonic structure induced by the Laplace-Beltrami operator.<n>We derive closed-form posterior distributions, optimal spectral truncation schemes, and sharp posterior contraction rates under integrated squared loss.
  arXiv  Detail & Related papers  (2026-01-28T12:10:33Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Multivariate root-n-consistent smoothing parameter free matching estimators and estimators of inverse density weighted expectations [51.000851088730684]
  We develop novel modifications of nearest-neighbor and matching estimators which converge at the parametric $sqrt n $-rate.<n>We stress that our estimators do not involve nonparametric function estimators and in particular do not rely on sample-size dependent parameters smoothing.
  arXiv  Detail & Related papers  (2024-07-11T13:28:34Z)
• Nonparametric regression on random geometric graphs sampled from submanifolds [2.3326951882644553]
  We analyze the frequentist behaviour of the posterior distribution arising from Bayesian priors designed through random basis expansion. We prove that the posterior contraction rates of such methods are minimax optimal (up to logarithmic factors) for any positive smoothness index.
  arXiv  Detail & Related papers  (2024-05-31T15:18:44Z)
• Sampling and estimation on manifolds using the Langevin diffusion [45.57801520690309]
  Two estimators of linear functionals of $mu_phi $ based on the discretized Markov process are considered.<n>Error bounds are derived for sampling and estimation using a discretization of an intrinsically defined Langevin diffusion.
  arXiv  Detail & Related papers  (2023-12-22T18:01:11Z)
• Randomized Physics-Informed Machine Learning for Uncertainty Quantification in High-Dimensional Inverse Problems [49.1574468325115]
  We propose a physics-informed machine learning method for uncertainty quantification in high-dimensional inverse problems. We show analytically and through comparison with Hamiltonian Monte Carlo that the rPICKLE posterior converges to the true posterior given by the Bayes rule.
  arXiv  Detail & Related papers  (2023-12-11T07:33:16Z)
• Intrinsic Bayesian Cramér-Rao Bound with an Application to Covariance Matrix Estimation [49.67011673289242]
  This paper presents a new performance bound for estimation problems where the parameter to estimate lies in a smooth manifold. It induces a geometry for the parameter manifold, as well as an intrinsic notion of the estimation error measure.
  arXiv  Detail & Related papers  (2023-11-08T15:17:13Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Convergence and concentration properties of constant step-size SGD through Markov chains [0.0]
  We consider the optimization of a smooth and strongly convex objective using constant step-size gradient descent (SGD) We show that, for unbiased gradient estimates with mildly controlled variance, the iteration converges to an invariant distribution in total variation distance. All our results are non-asymptotic and their consequences are discussed through a few applications.
  arXiv  Detail & Related papers  (2023-06-20T12:36:28Z)
• Kullback-Leibler and Renyi divergences in reproducing kernel Hilbert space and Gaussian process settings [0.0]
  We present formulations for regularized Kullback-Leibler and R'enyi divergences via the Alpha Log-Determinant (Log-Det) divergences. For characteristic kernels, the first setting leads to divergences between arbitrary Borel probability measures on a complete, separable metric space. We show that the Alpha Log-Det divergences are continuous in the Hilbert-Schmidt norm, which enables us to apply laws of large numbers for Hilbert space-valued random variables.
  arXiv  Detail & Related papers  (2022-07-18T06:40:46Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• The Connection between Discrete- and Continuous-Time Descriptions of Gaussian Continuous Processes [60.35125735474386]
  We show that discretizations yielding consistent estimators have the property of invariance under coarse-graining' This result explains why combining differencing schemes for derivatives reconstruction and local-in-time inference approaches does not work for time series analysis of second or higher order differential equations.
  arXiv  Detail & Related papers  (2021-01-16T17:11:02Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.