约束逆问题中泛函的基于优化的频率论置信区间：Burrus猜想的解决
Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture
AI总结：
本研究提出基于优化的约束逆问题泛函置信区间构建框架，将区间覆盖保证转化为第一类错误控制，反驳了Burrus猜想并给出反例，还通过数值示例验证框架并指明扩展方向。
摘要：
我们提出了一个基于优化的框架，用于构建约束逆问题中泛函的置信区间，确保有效的逐次频率论覆盖保证。我们的方法建立在现被称为严格界区间的方法之上，该方法最初由Burrus（1965）以及Rust和Burrus（1972）开创，提供了在推断过程中直接纳入任何关于参数的辅助信息而不引入外部偏差的途径。这类方法无需选择正则化先验，即可在不适定逆问题中实现不确定性量化。通过将基于优化的区间与约束似然比检验的反演相关联，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表征所得区间。在此过程中，我们反驳了Burrus猜想——该猜想假定，对于带正性约束的可能秩亏线性高斯模型，基于自由度为1的卡方分布分位数的校正足以在维持频率论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，并且我们利用随机占优论证构建了一个反例，我们还使用该反例证伪了该猜想的一般形式。我们通过若干数值示例说明了我们的框架，并为将Rust-Burrus方法扩展至带一般约束的非线性、非高斯场景提供了方向。
英文摘要：
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.
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