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.

翻译：我们证明了半参数Bernstein-von Mises定理，适用于具有独立先验的、含低维感兴趣参数与无穷维 nuisance 参数的部分线性回归模型。该结果缓解了因未知nuisance参数导致信息损失而产生的先验不变性条件。核心方法是对回归函数进行重新参数化，这类似于剖面似然的思想。因此，在变换后的模型中，由于不存在信息损失，先验不变性条件自动得到满足。鉴于这些先验稳定性条件可能对底层数据生成过程施加强限制，我们的结果相比原始参数化的部分线性模型，提供了更具鲁棒性的后验渐近正态性定理。