The geometry of uncertainty decomposition in profile-likelihood fits
Uncertainty decompositions in profile-likelihood fits are commonly reported through nuisance-parameter impacts, although shifting a fitted parameter and fluctuating the observation that constrains it answer different questions. Recently, Pinto et al. provided an explicit construction for uncertainty decomposition based on fluctuating the observations and argued that such a construction allows for a cleaner interpretation of systematic uncertainties in terms of physical sources. We provide a geometric description of the distinction between the two methods by using a coupled pair of fiber bundles. The geometrical approach clarifies the relationship between several methods traditionally used to estimate systematic uncertainties in high-energy physics. By describing the profiling as an information-orthogonal horizontal lift and the Schur complement as the induced metric on the parameter-of-interest manifold, physical-source uncertainties arise by mapping observation fluctuations to score covectors, raising them with the inverse total information, and pushing the resulting estimator covariance forward to the parameters of interest. This construction clarifies why nuisance-parameter impacts do not generally coincide with repeated-experiment source variances.