Systematic error modeling and nuisance marginalization

Let a primary reading y measure a physical value theta plus an unknown offset b, with independent Gaussian noise of known standard deviation sigma. An auxiliary calibration reading c measures that same b with known Gaussian standard deviation tau. In a frequentist analysis b is a fixed unknown parameter, while both y and c fluctuate in repetitions. Their likelihood factors only because the two noises are independent conditional on theta and b. If the calibration shares events with the primary data, that factorization needs revision.

Maximizing the joint likelihood gives the corrected estimate y minus c. Its repeated-sample variance is the sum of the two noise variances, because the correction contains the fluctuating calibration measurement with coefficient minus one. A profile likelihood holds theta fixed and maximizes over b. It retains nuisance dependence through its optimized value; it does not average b over a probability distribution. Calibration constraints are likelihood terms here, and should not automatically be called priors merely because their algebra is Gaussian.

For a Bayesian treatment, suppose an improper constant initial density for b on the real line is updated with the calibration alone. This gives a Gaussian distribution for b centered at c with variance tau squared. Use it once as the calibration-informed prior in the primary analysis. Integrating over b convolves two Gaussian distributions. With a flat prior for unrestricted theta, the resulting posterior is proper and Gaussian, centered at y minus c with variance sigma squared plus tau squared. The improper initial densities are acceptable here only because this posterior can be normalized.

Completing the square also shows that profiling yields the same theta-dependent quadratic as marginalization in this linear, constant-variance Gaussian case. The marginal integral contributes a theta-independent normalization. In nonlinear models, parameter-dependent widths, boundaries or non-Gaussian priors, that factor need not be constant and the two procedures can give different shapes. Neither integrating nor maximizing automatically provides frequentist coverage. Bayesian posterior content and a profile-based confidence construction require their respective interpretations and validation.

For several readings with the same offset, reducing the independent noise by averaging does not reduce the calibration uncertainty in the same way. Marginalizing a common offset induces off-diagonal covariance. Assigning a fresh independent offset to each reading is a different physical model and can understate the uncertainty of a common mean. The same principle applies to shared luminosity, energy scale and detector alignment parameters, although their sensitivities may differ and can depend on the physical parameters being fitted.

Uncertain efficiencies are bounded, positive scale factors are not unrestricted additive offsets, and theoretical shape alternatives may not be well described by one Gaussian parameter. Choose nuisance models that match calibration evidence and physical constraints, then examine sensitivity to defensible alternatives. Extra nuisance flexibility cannot guarantee protection against misspecification outside that family. Document what each parameter represents, which data constrain it, which correlations it creates, and whether calibration information is treated as a likelihood or a probability distribution over the parameter.