We introduce a density-power weighted variant for the Stein operator, called the $\gamma$-Stein operator. This is a novel class of operators derived from the $\gamma$-divergence, designed to build robust inference methods for unnormalized probability models. The operator's construction (weighting by the model density raised to a positive power $\gamma$ inherently down-weights the influence of outliers, providing a principled mechanism for robustness. Applying this operator yields a robust generalization of score matching that retains the crucial property of being independent of the model's normalizing constant. We extend this framework to develop two key applications: the $\gamma$-kernelized Stein discrepancy for robust goodness-of-fit testing, and $\gamma$-Stein variational gradient descent for robust Bayesian posterior approximation. Empirical results on contaminated Gaussian and quartic potential models show our methods significantly outperform standard baselines in both robustness and statistical efficiency.

翻译：我们引入了一种密度加权Stein算子的变体，称为$\\gamma$-Stein算子。这是一类从$\\gamma$-散度推导出的新型算子，旨在为未归一化概率模型构建鲁棒推断方法。该算子的构造（通过模型密度的正幂次$\\gamma$进行加权）本质上降低了异常值的影响，为鲁棒性提供了理论机制。应用该算子可得到分数匹配的鲁棒泛化形式，并保留了与模型归一化常数无关的关键性质。我们将此框架扩展至两个关键应用：用于鲁棒拟合优度检验的$\\gamma$-核化Stein差异，以及用于鲁棒贝叶斯后验近似的$\\gamma$-Stein变分梯度下降。在污染高斯模型和四次势能模型上的实证结果表明，我们的方法在鲁棒性和统计效率上均显著优于标准基线。