Profile likelihoods and nuisance parameters

For binned independent Poisson observations, each mean depends on the signal strength mu and nuisance vector theta. Auxiliary measurements such as control counts or calibration estimates supply additional likelihood factors. A Gaussian constraint can represent a measured calibration with an approximately Gaussian sampling distribution, but it should not be confused automatically with a Bayesian prior. The full model must state what is observed, what is fixed, and which variables would fluctuate in repeated experiments.

A nuisance parameter shared between channels creates a physical correlation. Giving every histogram bin an independent shift can remove that relationship, while forcing unrelated uncertainties to share a parameter can overconstrain them. Positive yields and efficiencies may require suitable parameterizations or explicit boundaries. A flexible likelihood can only quantify effects represented by its parameterized model; an omitted process or unsupported functional form is not repaired by the profiling operation itself.

Consider a signal-region count with mean mu s plus b and an independent control count with mean tau b. Here s and tau are known positive constants, while b is a background nuisance parameter. If the unconstrained fitted signal is positive, the global fit can reproduce both observed counts by setting b to the control count divided by tau and assigning the remaining signal-region excess to mu s. Under the no-signal hypothesis, the two regions instead jointly determine b.

This shift is precisely why a test should not generally fix b to its global best value while varying mu. The conditional maximum can use auxiliary information differently at each proposed hypothesis. Profiling maximizes rather than integrates; Bayesian marginalization instead integrates with a prior and answers a different inferential question. Neither procedure makes parameter degeneracies disappear. Weak auxiliary information can leave a broad profile even when the signal-region count is large.

A likelihood-ratio curve orders parameter values by fit quality, but converting it into a confidence interval requires a threshold with a justified sampling distribution. Wilks-type chi-square approximations need regularity, identifiability, sufficiently informative data, and appropriate treatment of boundaries. Small counts, nearly degenerate nuisance effects, or a parameter appearing only under an alternative can violate familiar asymptotic assumptions. Simulated repeated experiments can test or calibrate the chosen construction.

Frequentist coverage means that a specified procedure includes the fixed true parameter in the stated fraction of repeated datasets. It is not a posterior probability that the fixed parameter lies inside this already observed interval. Auxiliary fluctuations and the nuisance treatment belong to that procedure. Conditional or unconditional simulation choices can produce different questions, so a coverage study should state which parameters and observations vary. A successful fit and small optimizer error do not by themselves establish coverage or physical model adequacy.