Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems

Details Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems

2012-03-26

arxiv.org/abs/1203.5753v2

Mathematics

Statistics Theory

Scientific paper

We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors which are conjugate to the model so that the posterior distribution is also Gaussian. A method of identifying the posterior distribution using its precision operator is presented. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior consistency in a frequentist sense, and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. We show how these rates may be optimized by a choice of the scale parameter in the prior covariance operator. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator; more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known minimax rates of convergence in the case where the forward operator and the prior and noise covariances are all simultaneously diagonalizable, and confirm that the PDE method provides the same rates for a wide range of parameters. An elliptic PDE inverse problem is used to illustrate the power of the general theory.

Affiliated with

Also associated with

No associations

Rating

Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Posterior Consistency of the Bayesian Approach to Linear Ill-Posed Inverse Problems will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-642624