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Suppose an observable x is the measured value (negative or non-negative) of atrue mean mu (physically non-negative) in an experiment with a Gaussianresolution function with known fixed rms deviation s. The most powerfulone-sided upper confidence limit at 95% C.L. is UL = x+1.64s, which I refer toas the "original diagonal line". Perceived problems in HEP with small ornon-physical upper limits for x<0 historically led, for example, tosubstitution of max(0,x) for x, and eventually to abandonment in the ParticleData Group's Review of Particle Physics of this diagonal line relationshipbetween UL and x. Recently Cowan, Cranmer, Gross, and Vitells (CCGV) haveadvocated a concept of "power constraint" that when applied to this problemyields variants of diagonal line, including UL = max(-1,x)+1.64s. Thus it istimely to consider again what is problematic about the original diagonal line,and whether or not modifications cure these defects. In a 2002 Comment,statistician Leon Jay Gleser pointed to the literature on recognizable andrelevant subsets. For upper limits given by the original diagonal line, thesample space for x has recognizable relevant subsets in which the quoted 95%C.L. is known to be negatively biased (anti-conservative) by a finite amountfor all values of mu. This issue is at the heart of a dispute between JerzyNeyman and Sir Ronald Fisher over fifty years ago, the crux of which is therelevance of pre-data coverage probabilities when making post-data inferences.The literature describes illuminating connections to Bayesian statistics aswell. Methods such as that advocated by CCGV have 100% unconditional coveragefor certain values of mu and hence formally evade the traditional criteria fornegatively biased relevant subsets; I argue that concerns remain. Comparisonwith frequentist intervals advocated by Feldman and Cousins also sheds light onthe issues.

Negatively Biased Relevant Subsets Induced by the Most-Powerful One-Sided Upper Confidence Limits for a Bounded Physical Parameter