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Theory uncertainties on the inclusive cross section are typically not included in unfolded measurements. +The figure below shows a comparison of Likelihood based unfolding and a least-squares based unfolding as implemented in `RooUnfold`. + +/// details | **Show comparison** + +![](images/Unfolding_Comparison.png) + +/// + ## Regularization The main difference with respect to other models with multiple signal contributions is the introduction of **Regularization**, which is used to stabilize the unfolding process. @@ -124,9 +132,18 @@ Alternative, valid syntaxes are constr1 constr r_bin0+r_bin1 r_bin0,r_bin1 0.01 constr1 constr r_bin0+r_bin1 {r_bin0,r_bin1} delta[0.01] ``` + +The figure below shows an example unfolding using the "SVD regularization" approach with the least squares method (as implemented by `RooUnfold`) and implemented as a penalty term added to the likelihood using the maximum likelihood approach in `Combine`. + +/// details | **Show comparison** + +![](images/Unfolding_Regularization_comparison.png) + +/// + ### TUnfold method -The Tikhonov regularization as implemented in TUnfold uses the MC information, or rather the densities prediction, as a bias vector. +The Tikhonov regularization as implemented in `TUnfold` uses the MC information, or rather the densities prediction, as a bias vector. In order to give this information to Combine, a single datacard for each reco-level bin needs to be produced, so that we have access to the proper normalization terms during the minimization. In this case the bias vector is $\vec{x}_{obs}-\vec{x}_{true}$ Then one can write a constraint term in the datacard via (eg.)