We present an optimization-based framework to construct confidence intervals for functionals in constrained inverse problems, ensuring valid one-at-a-time frequentist coverage guarantees. Our approach builds upon the now-called strict bounds intervals, originally pioneered by Burrus (1965) and Rust and Burrus (1972), which offer ways to directly incorporate any side information about the parameters during inference without introducing external biases. This family of methods allows for uncertainty quantification in ill-posed inverse problems without needing to select a regularizing prior. By tying optimization-based intervals to an inversion of a constrained likelihood ratio test, we translate interval coverage guarantees into type I error control and characterize the resulting interval via solutions to optimization problems. Along the way, we refute the Burrus conjecture, which posited that, for possibly rank-deficient linear Gaussian models with positivity constraints, a correction based on the quantile of the chi-squared distribution with one degree of freedom suffices to shorten intervals while maintaining frequentist coverage guarantees. Our framework provides a novel approach to analyzing the conjecture, and we construct a counterexample employing a stochastic dominance argument, which we also use to disprove a general form of the conjecture. We illustrate our framework with several numerical examples and provide directions for extensions beyond the Rust-Burrus method for nonlinear, non-Gaussian settings with general constraints.

翻译：我们提出了一种基于优化的框架，用于在约束反问题中构建泛函的置信区间，确保具有有效的一次性频率覆盖保证。该方法基于如今被称为严格界限区间的方法，最初由Burrus（1965）以及Rust和Burrus（1972）开创，该方法允许在推断过程中直接纳入关于参数的任何辅助信息，而不会引入外部偏差。这一系列方法能够在无需选择正则化先验的情况下，对不适定反问题进行不确定性量化。通过将基于优化的区间与约束似然比检验的反演联系起来，我们将区间覆盖保证转化为第一类错误控制，并通过优化问题的解来刻画所得区间。在此过程中，我们反驳了伯鲁斯猜想，该猜想认为，对于可能秩亏且带有非负约束的线性高斯模型，基于自由度为1的卡方分布分位数进行修正，足以在保持频率覆盖保证的同时缩短区间。我们的框架为分析该猜想提供了一种新方法，并利用随机优势论证构造了一个反例，该论证也被用于否定该猜想的一般形式。我们通过多个数值示例展示了该框架，并提出了将Rust-Burrus方法推广到非线性、非高斯及一般约束情景的方向。