Regularizing Priors for linear inverse problems

This paper analyzes Bayesian estimation of functional parameters in econometric models that are characterized as the solution of a linear inverse problem. We focus on the leading examples in econometrics of instrumental regression and functional regression estimation. By using a Gaussian process prior distribution we propose the posterior mean as an estimator and study consistency, in the frequentist sense, of the posterior distribution. We show that the minimax rate of contraction can be obtained provided that either the regularity of the prior matches the regularity of the true parameter or the prior is scaled at an appropriate rate. The scaling parameter of the prior distribution plays the role of a regularization parameter. By using this fact we make clear that the posterior mean in a conjugate-Gaussian setting is equal to a Tikhonov type estimator in a frequentist setting. Finally, we propose an adaptive method for optimally selecting in practice the regularization parameter which is valid and meaningful also for a Tikhonov type estimator.