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On strong posterior contraction rates for Besov-Laplace priors in the white noise model

Emanuele Dolera, Stefano Favaro, Matteo Giordano

Abstract

In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior contraction rates, in strong Sobolev metrics, over the Besov spaces $B^β_{11}$, $β> d/2$, closing a gap in the existing literature. Our strong posterior contraction rates also imply that the posterior distributions arising from Besov-Laplace priors with matching regularity enjoy a desirable plug-in property for derivative estimation, entailing that the push-forward measures under differential operators optimally recover the derivatives of the unknown regression function. The proof of our results relies on the novel approach to posterior contraction rates, based on Wasserstein distance, recently developed by Dolera, Favaro and Mainini (Probability Theory and Related Fields, 2024). We show how this approach allows to overcome some technical challenges that emerge in the frequentist analysis of smoothness-matching Besov-Laplace priors.

On strong posterior contraction rates for Besov-Laplace priors in the white noise model

Abstract

In this article, we investigate the problem of estimating a spatially inhomogeneous function and its derivatives in the white noise model using Besov-Laplace priors. We show that smoothness-matching priors attains minimax optimal posterior contraction rates, in strong Sobolev metrics, over the Besov spaces , , closing a gap in the existing literature. Our strong posterior contraction rates also imply that the posterior distributions arising from Besov-Laplace priors with matching regularity enjoy a desirable plug-in property for derivative estimation, entailing that the push-forward measures under differential operators optimally recover the derivatives of the unknown regression function. The proof of our results relies on the novel approach to posterior contraction rates, based on Wasserstein distance, recently developed by Dolera, Favaro and Mainini (Probability Theory and Related Fields, 2024). We show how this approach allows to overcome some technical challenges that emerge in the frequentist analysis of smoothness-matching Besov-Laplace priors.

Paper Structure

This paper contains 19 sections, 6 theorems, 76 equations.

Table of Contents

  1. Introduction
  2. Existing literature and motivation
  3. Our contributions
  4. Organisation of the paper
  5. On the Wasserstein dynamics approach in the white noise model
  6. Preliminaries and notation
  7. The Gaussian white noise model, product priors and posteriors
  8. The Wasserstein dynamics approach to strong posterior contraction rates
  9. Main results
  10. Besov-Laplace priors
  11. Strong posterior contraction rates for Besov-Laplace priors
  12. Applications to derivative estimation
  13. Discussion
  14. Adaptation
  15. Beyond Besov-Laplace priors and the white noise model
  16. ...and 4 more sections

Key Result

Theorem 2

Assume observations $X^{(n)}\sim P^{(n)}_{f_0}$ from the white noise model Eq:WhiteNoise for some fixed $f_0\in B^\beta_1$, some $\beta> d/2$. Let $\Pi_\beta(\cdot|X^{(n)})$ be the posterior distribution arising from a $\beta$-regular Besov-Laplace prior, defined as in Eq:LaplPost with $\alpha=\beta as $n\to\infty$ for all positive real sequences $M_n\to\infty$.

Theorems & Definitions (14)

  • Remark 1: Extension to other basis and metrics
  • Theorem 2
  • proof
  • Remark 3: The case of $B^1_1$
  • Corollary 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • proof
  • ...and 4 more