Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems

By Sergios Agapiou, Stig Larsson, Andrew M. Stuart. 2012

96 citations recorded by the source corpus.

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/540fd3f311677333af1c0d32747c9924e7f9a5c4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. Kolyan Ray
2. Richard Nickl
3. Botond Szab'o
4. A. van der Vaart
5. Masoumeh Dashti
6. Andrew M. Stuart
7. B. Knapik
8. Judith Rousseau

In Statistics and Probability

1. Kolyan Ray
2. Richard Nickl
3. Botond Szab'o
4. A. van der Vaart
5. Judith Rousseau
6. Andrew M. Stuart
7. Masoumeh Dashti
8. Gabriel P. Paternain

Citations in the corpus

• Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, result): “In this regard our results are related to recent investigations of Bayesian inverse problems [29, 39, 4, 17, 28, 35], Bernstein-von Mises theorems [40, 32, 31, 33] and diffusion models [50, 38, 54, 34, 49, 26, 1].”
• The Bayesian Approach to Inverse Problems (2013, methodology): “r measure, finds application to problems from molecular dynamics in [83, 84]. Further material concerning the equivalence of posterior with respect to the prior may be found in [93, Chapters 3 and 6], [3], [4]. The equivalence of Gaussian measures is studied via the Feldman-H´ajeki theorem; see [29] ”
• Bayesian inverse problems with non-conjugate priors (2012, background): “[11, 13, 16, 29, 31] for some examples), but the situation of inverse problems has only recently been considered and then only in the conjugate setting [1, 20, 21], where explicit posterior expressions are available.”
• Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Regarding statistical (frequentist) properties of posterior measures, the case of linear G is again fairly well understood due to the explicit Gaussian structure of the posterior distribution, we refer here only to [35, 52, 2, 34, 39, 22, 26] and references therein.”
• Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, background): “Themore recent papers [22] and [1] study the inverse problemcase, but also obtain non-adaptive results only.”