We consider statistical inference problems under uncertain equality constraints, and provide asymptotically valid uncertainty estimates for inferred parameters. The proposed approach leverages the implicit function theorem and primal-dual optimality conditions for a particular problem class. The motivating application is multi-dimensional raking, where observations are adjusted to match marginals; for example, adjusting estimated deaths across race, county, and cause in order to match state all-race all-cause totals. We review raking from a convex optimization perspective, providing explicit primal-dual formulations, algorithms, and optimality conditions for a wide array of raking applications, which are then leveraged to obtain the uncertainty estimates. Empirical results show that the approach obtains, at the cost of a single solve, nearly the same uncertainty estimates as computationally intensive Monte Carlo techniques that pass thousands of observed and of marginal draws through the entire raking process.

翻译：本文研究在不确定等式约束下的统计推断问题，并为推断参数提供渐近有效的不确定性估计。所提出的方法针对特定问题类别，利用隐函数定理和原始-对偶最优性条件。该研究的应用背景是多维平衡法，即通过调整观测值以匹配边际分布；例如，为匹配州级全种族全死因总数，需按种族、县和死因调整估计死亡人数。我们从凸优化角度审视平衡法，为各类平衡应用提供显式的原始-对偶形式化描述、算法及最优性条件，进而利用这些条件获得不确定性估计。实证结果表明，该方法仅需单次求解即可获得与计算密集的蒙特卡洛技术几乎相同的不确定性估计结果，而蒙特卡洛技术需要将数千次观测值和边际抽样数据通过完整的平衡流程进行处理。