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Designing truncated priors for direct and inverse Bayesian problems

Sergios Agapiou, Peter Mathé

Abstract

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution in the frequentist setting. The present paper contributes to the area of Bayesian inverse problems by formulating a posterior contraction theory for linear inverse problems, with truncated Gaussian series priors, and under general smoothness assumptions. Emphasis is on the intrinsic role of the truncation point both for the direct as well as for the inverse problem, which are related through the modulus of continuity as this was recently highlighted by Knapik and Salomond (2018).

Designing truncated priors for direct and inverse Bayesian problems

Abstract

The Bayesian approach to inverse problems with functional unknowns, has received significant attention in recent years. An important component of the developing theory is the study of the asymptotic performance of the posterior distribution in the frequentist setting. The present paper contributes to the area of Bayesian inverse problems by formulating a posterior contraction theory for linear inverse problems, with truncated Gaussian series priors, and under general smoothness assumptions. Emphasis is on the intrinsic role of the truncation point both for the direct as well as for the inverse problem, which are related through the modulus of continuity as this was recently highlighted by Knapik and Salomond (2018).

Paper Structure

This paper contains 30 sections, 22 theorems, 147 equations.

Table of Contents

  1. Outline
  2. Setting and main result
  3. Notation
  4. Prior distribution and posterior contraction
  5. Relating operators in Hilbert space
  6. Smoothness concept
  7. Main result
  8. Direct signal estimation under truncated Gaussian priors
  9. Native and inherited Gaussian priors
  10. Basic SPC bound
  11. Optimized SPC bound
  12. Interlude
  13. Modulus of continuity for inverse problems
  14. Modulus of continuity
  15. Bounding the modulus of continuity
  16. ...and 15 more sections

Key Result

Proposition 2.1

Assume we put a, truncated at level $k_n$, Gaussian prior on $f$. Let $\delta_n\to0$ be a rate of contraction for the direct problem eq:direct around $g_0=~A f_0\in~Y$, for some $f_0\in X$. Then $\varepsilon_n:=\omega_{f_0}(H^{-1/2},X_{k_{n}},\delta_n)$, where $H=A^\ast A$, is a rate of contraction

Theorems & Definitions (58)

  • Example : $\alpha$-regular prior
  • Definition 1: index function
  • Remark 1
  • Definition 2: Source set
  • Remark 2
  • Example : Sobolev-type smoothness
  • Proposition 2.1
  • Theorem 1
  • Proposition 3.1
  • Proposition 3.2: MR1062717
  • ...and 48 more