- Thesis:
- Master in Mathematics
- Author:
- Stephan Bischofberger
- Title:
- Posterior contraction for Gaussian inverse problems in Hilbert scales
- Supervisor:
- Jan JOHANNES
- Abstract:
- We consider mildly ill-posed inverse problems in separable Hilbert spaces with Gaussian noise equipped with a Gaussian prior. Under the frequentistic assumption of a true parameter associated with the data-generating process, in Hilbert scales it is shown that the posterior contracts to that parameter. In particular, this gener- alizes findings in the Gaussian sequence model obtained by Knapik, van der Vaart and van Zanten (2011). The representation of the inverse problem as a sequence model is essentially based on the fact that the spectral decomposition of the under- lying linear operator is known. In particular, in this representation, the regularity conditions imposed on the solution of the inverse problem are expressed in terms of the eigenbasis of the operator. The considered generalization allows us to sepa- rate the regularity conditions on the solution and the spectral representation of the operator. Under additional mild assumptions, we show that the contraction rates in Hilbert scales and in the Gaussian sequence model coincide. Furthermore, we investigate the contraction of the posterior distribution of a lin- ear functional applied to the parameter. It is further shown that similar results can be achieved with a Gaussian sieve prior. We give sufficient conditions on the con- sidered prior distributions, such that the contraction of the posterior attains the nonparametric minimax-optimal rate for an optimal choice of the prior parameters. A simulation study for the Volterra operator illustrates the asymptotic results.
- Reference:
- B. T. Knapik, A. W. van der Vaart and J. H. van Zanten. Bayesian inverse problems with Gaussian priors. The Annals of Statistics, 39(5):2626-2657, 2011.
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