To infer a function value on a specific point $x$, it is essential to assign higher weights to the points closer to $x$, which is called local polynomial / multivariable regression. In many practical cases, a limited sample size may ruin this method, but such conditions can be improved by the Prediction-Powered Inference (PPI) technique. This paper introduced a specific algorithm for local multivariable regression using PPI, which can significantly reduce the variance of estimations without enlarge the error. The confidence intervals, bias correction, and coverage probabilities are analyzed and proved the correctness and superiority of our algorithm. Numerical simulation and real-data experiments are applied and show these conclusions. Another contribution compared to PPI is the theoretical computation efficiency and explainability by taking into account the dependency of the dependent variable.

翻译：为推断特定点$x$上的函数值，必须赋予靠近$x$的点更高权重，这种方法称为局部多项式/多元回归。在许多实际场景中，有限的样本量可能破坏该方法的有效性，但预测驱动推断技术能够改善此类状况。本文提出了一种基于PPI的局部多元回归算法，该算法可在不增大误差的前提下显著降低估计方差。通过对置信区间、偏差校正和覆盖概率的分析，证明了所提算法的正确性与优越性。数值模拟与真实数据实验验证了上述结论。相较于原始PPI方法，本研究的另一贡献在于通过考虑因变量的依赖性，实现了理论计算效率的提升与可解释性的增强。