The semi-parametric Bernstein-von Mises theorem for regression models with symmetric errors

In a smooth semi-parametric model, the marginal posterior distribution of a finite-dimensional parameter of interest is expected to be asymptotically equivalent to the sampling distribution of any efficient point estimator. This assertion leads to asymptotic equivalence of the credible and confidence sets of the parameter of interest, and is known as the semi-parametric Bernstein-von Mises theorem. In recent years, this theorem has received much attention and has been widely applied. Here, we consider models in which errors with symmetric densities play a role. Specifically, we show that the marginal posterior distributions of the regression coefficients in linear regression and linear mixed-effect models satisfy the semi-parametric Bernstein-von Mises assertion. As a result, Bayes estimators in these models achieve frequentist inferential optimality, as expressed, for example, in Hájek’s convolution and asymptotic minimax theorems. For the prior on the space of error densities, we provide two well-known examples, namely, the Dirichlet process mixture of normal densities and random series priors. The results provide efficient estimates of the regression coefficients in the linear mixed-effect model, for which no efficient point estimators currently exist.