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); Rust and Burrus (1972), which offer ways to directly incorporate any side information about parameters during inference without introducing external biases. Notably, this family of methods allows for uncertainty quantification in ill-posed inverse problems without needing to select a regularizing prior. By tying our proposed 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 of 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 analyze the conjecture and construct a counterexample by 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 non-linear, 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造泛函的置信区间，确保具有有效的逐次频率覆盖保证。该方法建立在如今被称为严格界限区间（最初由Burrus（1965）以及Rust和Burrus（1972）开创）的基础上，该区间能够在不引入外部偏差的情况下，在推断过程中直接纳入关于参数的任意辅助信息。值得注意的是，这类方法无需选择正则化先验，即可对不适定反问题进行不确定性量化。通过将我们提出的区间与约束似然比检验的反演相联系，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，我们否定了Burrus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校正，足以在维持频率覆盖保证的前提下缩短区间。我们的框架提供了一种分析该猜想的新方法，并通过随机占优论证构造了反例，该方法同样用于否证该猜想的广义形式。我们通过多个数值算例展示了所提框架，并针对具有一般约束的非线性、非高斯情形，给出了超出Rust-Burrus方法的扩展方向。