Asymptotic normality of semiparametric and nonparametric posterior distributions

In this article it is shown that the marginal semiparametric and nonparametric posterior distributions for a parameter of interest behave like an ordinary parametric posterior distribution. This in practice provides support of the utility of marginal semiparametric and nonparametric posterior distributions. In particular, the marginal semiparametric and nonparametric posterior distributions are asymptotically normal and centered at the corresponding maximum likelihood estimates (MLEs) or posterior means, with covariance matrix the inverse of the Fisher information. Additionally, the semiparametric and nonparametric MLEs for the parameter of interest and the marginal posterior means are asymptotically normal and centered at the true parameter, with the same covariance matrix. The results are a semiparametric version and a nonparametric version of the parametric Bayesian central limit theorem that establish a connection between the semiparametric and the nonparametric Bayesian inference and their frequentist counterparts.