Posterior consistency of the Bayesian approach to linear ill-posed inverse problems
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying the posterior distribution using its precision operator is presented. Work- ing with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior con- sistency 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 diagonaliz- able, 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.