# Bayesian posterior contraction rates for linear severely ill-posed inverse problems

@article{Agapiou2013BayesianPC, title={Bayesian posterior contraction rates for linear severely ill-posed inverse problems}, author={Sergios Agapiou and Andrew M. Stuart and Yuan-Xiang Zhang}, journal={Journal of Inverse and Ill-posed Problems}, year={2013}, volume={22}, pages={297 - 321} }

Abstract. We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We…

## 34 Citations

### Posterior Contraction in Bayesian Inverse Problems Under Gaussian Priors

- Mathematics, Computer Science
- 2018

This work reviews and re-derive several existing results, and establishes minimax contraction rates in cases which have not been considered until now, and shows how to overcome saturation in an empirical Bayesian framework by using a non-centered data-dependent prior.

### Consistency of the posterior distribution in generalized linear inverse problems

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The approach allows to obtain and, using preconditioning improve after saturation, minimax rates of contraction established in previous studies, and establishes minimax contraction rates in cases which have not been considered so far.

### Some results on contraction rates for Bayesian inverse problems

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We prove a general lemma for deriving contraction rates for linear inverse problems with non parametric nonconjugate priors. We then apply it to Gaussian priors and obtain minimax rates in mildly…

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Posterior consistency justifies the use of the Bayesian approach very much in the same way as error bounds and convergence results for regularization techniques do and it is then of interest to show that the resulting sequence of posterior measures arising from this sequence of data concentrates around the truth used to generate the data.

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It is proved that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view, and frequentist guarantees for certain credible balls centred at $\bar{f}$ are derived.

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In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior distribution for nonparametric inverse problems. We present a general theorem that allows…

### Posterior consistency and convergence rates for Bayesian inversion with hypoelliptic operators

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The Bayesian approach to inverse problems is studied in the case where the forward map is a linear hypoelliptic pseudodifferential operator and measurement error is additive white Gaussian noise. The…

### On the Bernstein-Von Mises Theorem for High Dimensional Nonlinear Bayesian Inverse Problems

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We prove a Bernstein-von Mises theorem for a general class of high dimensional nonlinear Bayesian inverse problems in the vanishing noise limit. We propose a sufficient condition on the growth rate…

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