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Bayesian inverse problems with partial observations. (English) Zbl 1422.62139

Summary: We study a nonparametric Bayesian approach to linear inverse problems under discrete observations. We use the discrete Fourier transform to convert our model into a truncated Gaussian sequence model, that is closely related to the classical Gaussian sequence model. Upon placing the truncated series prior on the unknown parameter, we show that as the number of observations \(n \rightarrow \infty\), the corresponding posterior distribution contracts around the true parameter at a rate depending on the smoothness of the true parameter and the prior, and the ill-posedness degree of the problem. Correct combinations of these values lead to optimal posterior contraction rates (up to logarithmic factors). Similarly, the frequentist coverage of Bayesian credible sets is shown to be dependent on a combination of smoothness of the true parameter and the prior, and the ill-posedness of the problem. Oversmoothing priors lead to zero coverage, while undersmoothing priors produce highly conservative results. Finally, we illustrate our theoretical results by numerical examples.

MSC:

62G07 Density estimation
60G15 Gaussian processes
42B05 Fourier series and coefficients in several variables
35K05 Heat equation
65C60 Computational problems in statistics (MSC2010)
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