Parametrization, Prior Independence, and Posterior Asymptotic Normality in the Partially Linear Model

We prove a semiparametric Bernstein-von Mises theorem for a partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. Our result mitigates a prior invariance condition that arises from a loss of information in not knowing the nuisance parameter. The key device is a reparametrization of the regression function that is in the spirit of profile likelihood, and, as a result, the prior invariance condition is automatically satisfied because there is no loss of information in the transformed model. As these prior stability conditions can impose strong restrictions on the underlying data-generating process, our results provide a more robust posterior asymptotic normality theorem than the original parametrization of the partially linear model.