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SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

Enric Alberola-Boloix, Ioar Casado-Telletxea

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.

SPDE Methods for Nonparametric Bayesian Posterior Contraction and Laplace Approximation

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.
Paper Structure (20 sections, 13 theorems, 126 equations)

This paper contains 20 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Contributions and structure of the paper
  3. Related work
  4. Background
  5. Bayesian inference and posterior contraction
  6. Infinite-dimensional Langevin equation
  7. Posterior contraction rates (PCRs)
  8. PCRs under strong concavity
  9. PCRs under weak concavity
  10. Laplace-type Approximation
  11. Necessary conditions for measure comparison
  12. Smoothness and Empirical Process bounds on $H^{*}$
  13. Consistency assumptions
  14. Laplace Approximation
  15. Example: Linear Gaussian model
  16. ...and 5 more sections

Key Result

Proposition 2.2

Consider a sequence of probability measures $(\pi_t)_{t\geq0}$ on $\mathcal{H}$ such that $\pi_t\Rightarrow\pi^*$ (weak convergence), and suppose that we have $\sup_{t\geq 0} \mathbb{E}_{\pi_t}\left[\|X\|_\mathcal{H}^r \right]<\infty$ and $\mathbb{E}_{\pi^*}\left[\|X\|_\mathcal{H}^r \right]<\infty$

Theorems & Definitions (27)

  • Definition 2.1
  • Proposition 2.2
  • Theorem 3.1
  • Theorem 3.2
  • Remark 1
  • Theorem 4.1
  • Remark 2
  • Remark 3
  • Theorem 4.2
  • Corollary 5.1
  • ...and 17 more