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Efficient Bayesian Inference in Strictly Semi-parametric Linear Inverse Problems

Adel Magra, Aad van der Vaart

TL;DR

This work develops a rigorous semi-parametric Bayesian framework for inverse problems where the forward map $K_\theta$ is linear in an infinite-dimensional signal $f$ but the operator depends on a scalar parameter $\theta$. By combining LAN expansions, least favourable directions, and posterior contraction results with carefully chosen priors (notably re-scaled Gaussian process priors), the authors prove Bernstein–von Mises theorems for the marginal posterior of $\theta$, enabling asymptotically optimal uncertainty quantification. They establish regularity, identifiability, and contraction conditions in abstract form, then instantiate them in two concrete applications: semi-blind deconvolution and attenuated X-ray transforms, including both symmetric/zero-location deconvolution models and constant attenuation X-ray settings. The results yield explicit rates and conditions under which the $\theta$-posterior is asymptotically Gaussian with a variance given by the efficient information, highlighting how ill-posedness can aid parametric inference in this mixed parametric-nonparametric regime. Practical impact lies in providing principled, computable uncertainty quantification for location and attenuation parameters in imaging-type inverse problems, with guidance on priors and regularity needed to achieve BvM behavior.

Abstract

We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal $f$ corrupted by statistical noise, with the transformation $K_θ$ being linear but unknown up to a scalar $θ$. We adopt a Bayesian approach and put a prior on the pair $(θ,f)$ and prove a Bernstein-von Mises theorem for the marginal posterior of $θ$ under regularity conditions on the operators $K_θ$ and on the prior. We apply our results to the recovery of location parameters in semi-blind deconvolution problems and to the recovery of attenuation constants in X-ray tomography.

Efficient Bayesian Inference in Strictly Semi-parametric Linear Inverse Problems

TL;DR

This work develops a rigorous semi-parametric Bayesian framework for inverse problems where the forward map is linear in an infinite-dimensional signal but the operator depends on a scalar parameter . By combining LAN expansions, least favourable directions, and posterior contraction results with carefully chosen priors (notably re-scaled Gaussian process priors), the authors prove Bernstein–von Mises theorems for the marginal posterior of , enabling asymptotically optimal uncertainty quantification. They establish regularity, identifiability, and contraction conditions in abstract form, then instantiate them in two concrete applications: semi-blind deconvolution and attenuated X-ray transforms, including both symmetric/zero-location deconvolution models and constant attenuation X-ray settings. The results yield explicit rates and conditions under which the -posterior is asymptotically Gaussian with a variance given by the efficient information, highlighting how ill-posedness can aid parametric inference in this mixed parametric-nonparametric regime. Practical impact lies in providing principled, computable uncertainty quantification for location and attenuation parameters in imaging-type inverse problems, with guidance on priors and regularity needed to achieve BvM behavior.

Abstract

We consider the efficient inference of finite dimensional parameters arising in the context of inverse problems. Our setup is the observation of a transformation of an unknown infinite dimensional signal corrupted by statistical noise, with the transformation being linear but unknown up to a scalar . We adopt a Bayesian approach and put a prior on the pair and prove a Bernstein-von Mises theorem for the marginal posterior of under regularity conditions on the operators and on the prior. We apply our results to the recovery of location parameters in semi-blind deconvolution problems and to the recovery of attenuation constants in X-ray tomography.
Paper Structure (18 sections, 14 theorems, 143 equations)

This paper contains 18 sections, 14 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Model Identifiability and Information
  4. A first BvM
  5. Re-scaled Gaussian Process Priors
  6. Posterior Contraction with re-scaled Gaussian Priors
  7. A BvM with re-scaled Gaussian Priors
  8. Application to Semi-blind Deconvolution
  9. Regularity and Stability Assumptions
  10. BvM for Semi-blind Deconvolution
  11. Application to Attenuated X-ray Transforms
  12. Regularity and Stability Assumptions
  13. BvM for Constantly Attenuated X-ray transforms
  14. Proofs
  15. Proof of Theorem \ref{['thm: main']}
  16. ...and 3 more sections

Key Result

Theorem 2.2

Let $(\theta_0,f_0)\in\Theta\times H$ be such that $\tilde{i}_{\theta_0,f_0}>0$. Suppose Assumption asspt1 is satisfied. Assume that for $M$ large enough, the posterior mass of $\{f: \|f\|_{H}<M\}$ tends to 1. Assume further that the posterior is consistent at $(\theta_0,f_0)$ w.r.t the semi-metric Then the BvM holds at $(\theta_0,f_0)$.

Theorems & Definitions (26)

  • Theorem 2.2
  • Corollary 2.3
  • proof
  • Theorem 3.3: Contraction in the natural testing metric
  • Corollary 3.4: Decoupled Contraction and in the Derivative
  • proof
  • Theorem 3.5
  • proof
  • Lemma 4.1
  • Proposition 4.2
  • ...and 16 more