We consider a class of linear ill-posed inverse problems arising from inversion of a compact operator with singular values which decay exponentially to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown function. The observational noise is assumed to be Gaussian; as a consequence the prior is conjugate to the likelihood so that the posterior distribution is also Gaussian. We study Bayesian posterior consistency in the small observational noise limit. We assume that the forward operator and the prior and noise covariance operators commute with one another. We show how, for given smoothness assumptions on the truth, the scale parameter of the prior, which is a constant multiplier of the prior covariance operator, can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.
Funding source: China Scholarship Council
Funding source: NNSF of China
Award Identifier / Grant number: 11171136
Funding source: ERC
The three authors would like to thank Bartek Knapik and Harry van Zanten for helpful discussions. The authors are also grateful to two referees for several very useful comments.
© 2014 by Walter de Gruyter Berlin/Boston