Check here for a new paper by Dong Yan, Aad van der Vaart and me on Bayesian inverse problems with partial observations. Dong is our PhD student (now working at Nationale-Nederlanden), and in fact this is his first published work. Congratulations!

The statistical literature on linear inverse problems usually assumes availability of a complete record of observations. Here, however, we deal with the case of 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 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. Similarly, we show that the frequentist coverage of Bayesian credible sets is dependent on a combination of smoothness of the true parameter and the prior, and the ill-posedness of the problem. Finally, some numerical examples illustrate our theoretical findings.