Regularizing Priors for Linear Inverse Problems

This paper proposes a new Bayesian approach for estimating, nonparametrically, parameters in econometric models that are characterized as the solution of a linear inverse problem. By using a Gaussian process prior distribution we propose the posterior mean as an estimator and prove consistency, in the frequentist sense, of the posterior distribution. Consistency of the posterior distribution provides a frequentist validation of our Bayesian procedure. We show that the minimax rate of contraction of the posterior distribution 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. We propose a new, and easy-to-implement, data-driven method for optimally selecting in practice this regularization parameter. Moreover, we make clear that the posterior mean, in a conjugate-Gaussian setting, is equal to a Tikhonov-type estimator in a frequentist setting so that our data-driven method can be used in frequentist estimation as well. Finally, we apply our general methodology to two leading examples in econometrics: instrumental regression and functional regression estimation.