Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations

Journal article (2024)

Authors

Dong Yan Student
S Gugushvili Wageningen University & Research
A.W. van der Vaart Statistics - EEMCS
Research Group
Statistics (EEMCS) (TU Delft)
Copyright: © 2024 Dong Yan, Shota Gugushvili, A.W. van der Vaart
DOI: https://doi.org/10.1007/s13171-024-00342-0
Regression Interpolation Posterior contraction rate Galerkin Nonparametric Bayesian estimation 35R30 62G20 Adaptive estimation Fixed design Gaussian prior Hilbert scale Linear inverse problem Random series prior Regularity scale
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Published Date

2024

Language

English

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Faculty

Electrical Engineering, Mathematics and Computer Science

Department

Delft Institute of Applied Mathematics

Research Group

Statistics

Abstract

We obtain rates of contraction of posterior distributions in inverse problems with discrete observations. In a general setting of smoothness scales we derive abstract results for general priors, with contraction rates determined by discrete Galerkin approximation. The rate depends on the amount of prior concentration near the true function and the prior mass of functions with inferior Galerkin approximation. We apply the general result to non-conjugate series priors, showing that these priors give near optimal and adaptive recovery in some generality, Gaussian priors, and mixtures of Gaussian priors, where the latter are also shown to be near optimal and adaptive.