SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation
This work addresses theoretical challenges in nonparametric Bayesian inference, providing rigorous contraction rates and approximations for posterior distributions, which is incremental as it builds on existing diffusion-based methods.
The authors tackled the problem of deriving posterior contraction rates and finite-sample Bernstein von Mises results for nonparametric Bayesian models by extending a diffusion-based framework to infinite dimensions, representing the posterior as an invariant measure of a Langevin SPDE to control moments and obtain non-asymptotic concentration rates in Hilbert norms, and establishing a quantitative Laplace approximation, illustrated in a nonparametric linear Gaussian inverse problem.
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.