This paper aims at developing a quasi-Bayesian analysis of the nonparametric instrumental variables model, with a focus on the asymptotic properties of quasi-posterior distributions. In this paper, instead of assuming a distributional assumption on the data generating process, we consider a quasi-likelihood induced from the conditional moment restriction, and put priors on the function-valued parameter. We call the resulting posterior quasi-posterior, which corresponds to “Gibbs posterior” in the literature. Here we focus on priors constructed on slowly growing finite-dimensional sieves. We derive rates of contraction and a nonparametric Bernstein–von Mises type result for the quasi-posterior distribution, and rates of convergence for the quasi-Bayes estimator defined by the posterior expectation. We show that, with priors suitably chosen, the quasi-posterior distribution (the quasi-Bayes estimator) attains the minimax optimal rate of contraction (convergence, resp.). These results greatly sharpen the previous related work.
"Quasi-Bayesian analysis of nonparametric instrumental variables models." Ann. Statist. 41 (5) 2359 - 2390, October 2013. https://doi.org/10.1214/13-AOS1150