In this paper, we elucidate the geometry of Stein's method of moments (SMoM). SMoM is a parameter estimation method based on the Stein operator, and yields a wide class of estimators that do not depend on the normalizing constant. We present a canonical decomposition of an SMoM estimator after centering the score matching estimator, which sheds light on the central role of the score matching within the SMoM framework. Using this decomposition, we construct an SMoM estimator that improves upon the score matching estimator in the asymptotic variance. We also discuss the connection between SMoM and the Wasserstein geometry. Specifically, using the Wasserstein score function, we provide a geometrical interpretation of the gap in the asymptotic variance between the score matching estimator and the maximum likelihood estimator. Furthermore, it is shown that the score matching estimator is asymptotically efficient if and only if the Fisher score functions span the same space as the Wasserstein score functions.

翻译：本文阐明了Stein矩量法的几何结构。Stein矩量法是一种基于Stein算子的参数估计方法，可产生一类不依赖于归一化常数的估计量。我们提出了在中心化得分匹配估计量后Stein矩量法估计量的典型分解，这揭示了得分匹配在Stein矩量法框架中的核心作用。利用该分解，我们构建了一个在渐近方差上优于得分匹配估计量的Stein矩量法估计量。我们还讨论了Stein矩量法与Wasserstein几何之间的联系。具体而言，利用Wasserstein得分函数，我们给出了得分匹配估计量与最大似然估计量之间渐近方差差异的几何解释。此外，研究证明当且仅当Fisher得分函数与Wasserstein得分函数张成相同空间时，得分匹配估计量才是渐近有效的。