Bayesian Posterior Contraction Rates for Linear Severely Illposed Inverse Problems
Abstract
We consider a class of linear illposed 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. If the observational noise is assumed to be Gaussian then this 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 can be adjusted to optimize the rate of posterior contraction to the truth, and we explicitly compute the logarithmic rate.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.1563
 Bibcode:
 2012arXiv1210.1563A
 Keywords:

 Mathematics  Statistics Theory;
 62G20;
 62C10;
 35R30;
 45Q05
 EPrint:
 25 pages, 2 figures