Parametrization, Prior Independence, and the Semiparametric Bernstein-von Mises Theorem for the Partially Linear Model

We prove a semiparametric Bernstein-von Mises theorem for a homoskedastic partially linear regression model with independent priors for the low-dimensional parameter of interest and the infinite-dimensional nuisance parameters. Our result avoids a prior invariance condition that arises from a loss of information in not knowing the nuisance parameter. The key idea is a feasible reparametrization of the regression function that mimics the Gaussian profile likelihood. Such a device allows a researcher to assume independent priors for the model parameters while automatically accounting for loss of information associated with not knowing the nuisance parameter. As these prior stability conditions often impose strong restrictions on the underlying data-generating process, our results provide a more robust asymptotic normality theorem than the original parametrization of the partially linear model.