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We present an optimization-based framework to construct confidence intervalsfor functionals in constrained inverse problems, ensuring valid one-at-a-timefrequentist coverage guarantees. Our approach builds upon the now-called strictbounds intervals, originally pioneered by Burrus (1965); Rust and Burrus(1972), which offer ways to directly incorporate any side information aboutparameters during inference without introducing external biases. Notably, thisfamily of methods allows for uncertainty quantification in ill-posed inverseproblems without needing to select a regularizing prior. By tying our proposedintervals to an inversion of a constrained likelihood ratio test, we translateinterval coverage guarantees into type-I error control, and characterize theresulting interval via solutions of optimization problems. Along the way, werefute the Burrus conjecture, which posited that, for possibly rank-deficientlinear Gaussian models with positivity constraints, a correction based on thequantile of the chi-squared distribution with one degree of freedom suffices toshorten intervals while maintaining frequentist coverage guarantees. Ourframework provides a novel approach to analyze the conjecture and construct acounterexample by employing a stochastic dominance argument, which we also useto disprove a general form of the conjecture. We illustrate our framework withseveral numerical examples and provide directions for extensions beyond theRust-Burrus method for non-linear, non-Gaussian settings with generalconstraints.

Optimization-based frequentist confidence intervals for functionals in constrained inverse problems: Resolving the Burrus conjecture