Causal effect estimation often succeeds cost-constrained sequential data collection. This work considers multivariate linear front-door models with arbitrary unobserved confounding on treatment and response. We optimize the experimental design by balancing the statistical efficiency and measurement costs through partial data. The full-data efficient influence function for the causal effect is derived, together with the geometry of all observed-data influence functions. This characterization yields a closed-form optimal sampling policy and an estimator to minimize the asymptotic variance of regular asymptotically linear (RAL) estimators within a class of augmented full-data influence functions. The resulting design also covers back-door estimation. In simulations and applications to biological, medical, and industrial datasets, the optimized designs achieve substantial efficiency gains ($5.3\%$ to $31.9\%$) over naive full-sampling strategies.

翻译：因果效应估计通常需要在成本约束下进行顺序数据收集。本文考虑具有任意未观测混杂因素（作用于处理和响应）的多元线性前门模型。我们通过优化实验设计，利用部分数据平衡统计效率与测量成本。推导出了因果效应的全数据有效影响函数，以及所有观测数据影响函数的几何结构。这一刻画给出了一个闭式最优采样策略和估计量，用于在增强全数据影响函数类中最小化正则渐近线性（RAL）估计量的渐近方差。所得设计也覆盖了后门估计。在生物、医学和工业数据集的模拟与应用中，优化设计相比朴素全采样策略实现了显著的效率提升（5.3%至31.9%）。