Asymptotic properties of nonparametric estimation and quantile regression in Bayesian structural equation models

We study the asymptotic properties of nonparametric Bayesian structural equation models (SEMs). Under mild conditions, when adjusting nonparametric error distributions, the posteriors of Bayesian SEMs achieve the optimal convergence rate up to logn terms in the nonparametric means and nonlinear relationships of the latent variables. Furthermore, we consider quantile regressions of the error and latent variables in Bayesian SEMs, and we show posterior consistency in Bayesian quantile regression. The theoretical results are validated using simulation studies.