Bayesian Linear Inverse Problems in Regularity Scales with Discrete Observations

Dong Yan, Shota Gugushvili () and Aad Vaart ()
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Dong Yan: TU Delft
Shota Gugushvili: Wageningen University & Research
Aad Vaart: TU Delft

Sankhya A: The Indian Journal of Statistics, 2024, vol. 86, issue 1, No 13, 228-254

Abstract: 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.

Keywords: Adaptive estimation; Gaussian prior; Hilbert scale; Linear inverse problem; Nonparametric Bayesian estimation; Posterior contraction rate; Random series prior; Regularity scale; Regression; Fixed design; Interpolation; Galerkin; 62G20; 35R30 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13171-024-00342-0

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