Post-hoc regularisation of unfolded cross-section measurements

Neutrino cross-section measurements are often presented as unfolded binned distributions in ``true'' variables. The ill-posedness of the unfolding problem can lead to results with strong anti-correlations and fluctuations between bins, which make comparisons to theoretical models in plots difficult. To alleviate this problem, one can introduce regularisation terms in the unfolding procedure. These suppress potential anti-correlations in the result, at the cost of introducing some bias towards the expected shape of the result, like the smoothness of bin-to-bin differences. Using simple linear algebra, it is possible to regularise any result that is presented as a central value and a covariance matrix. This ``post-hoc'' regularisation is generally much faster than repeating the unfolding method with different regularisation terms. The method also yields a regularisation matrix $A$ which connects the regularised to the unregularised result, and can be used to retain the full statistical power of the unregularised result when publishing a nicer looking regularised result. When doing this, the bias of the regularisation can be understood as a data visualisation problem rather than a statistical one. The strength of the regularisation can be chosen by minimising the difference between the implicitly uncorrelated distribution shown in the plots and the actual distribution described by the unregularised central value and covariance. The Wasserstein distance is a suitable measure for that difference. Aside from minimising the difference between the shown and the actual result, additional information can be provided by showing the local log-likelihood gradient of the models shown in the plots. This adds more information about where the model is ``pulled'' by the data than just comparing the bin values to the data's central values.