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.

翻译：我们证明了在同方差部分线性回归模型中，当感兴趣的低维参数与无穷维 nuisance 参数采用独立先验时，半参数Bernstein-von Mises定理成立。该结果规避了因未知 nuisance 参数导致信息损失而产生的先验不变性条件。核心思想是对回归函数进行可行的重新参数化，以模仿高斯剖面似然函数。这一方法允许研究者对模型参数设定独立先验，同时自动考虑未知 nuisance 参数带来的信息损失。由于此类先验稳定性条件通常对底层数据生成过程施加严格约束，我们的结果相比部分线性模型的原始参数化形式，提供了更具鲁棒性的渐近正态性定理。