Florens, Jean-Pierre and Simoni, Anna (2013) Regularizing Priors for Linear Inverse Problems. TSE Working Paper, n. 13-384

Abstract

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.