  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.
