Description
Two-point uncertainties
Two-point uncertainties are built via a comparison of a nominal sample with another one, for example one that was simulated using a different method.
In HistFactory, such an uncertainty is implemented via two correlated components, an OverallSys for the normalization effect and a HistoSys for the bin-by-bin shape variations.
The templates for the HistoSys are normalized such that their total yield per region matches that of the nominal sample. The normalization effect is factored out into the OverallSys component.
Two-point uncertainties are usually symmetrized, for example by defining the alternative sample as the "up" variation with respect to the nominal distribution, and mirroring it to get the "down" variation in the opposite direction.
In the implementation, it matters whether the normalization needed for the HistoSys happens before or after symmetrization.
Is there a recommended order, or at least a commonly implemented one?
Implementation example
Take a nominal two-bin histogram [10, 12] and an “up” variation [14, 14], which should be symmetrized and implemented in a HistFactory workspace.
For the normalization component, an OverallSys with up/down factors of (14+14)/(10+12) = 1.2727... and 2 - 1.2727... = 0.7272... is needed. There are two ways to obtain the HistoSys templates.
Method 1 (normalize -> symmetrize), which is what cabinetry implements
Re-normalize the "up" variation to have the same yield as nominal, to a template of [11, 11]. The HistoSys "up" template is [11, 11], and the symmetrized "down" template is obtained as twice the nominal minus the normalized "up" variation, as 2*[10, 12] - [11, 11] = [9, 13]. The resulting templates are:
"up": [11, 11]
"down": [9, 13]
Method 2 (symmetrize -> normalize)
The distance of the un-normalized "up" variation to the nominal template is [14, 14] - [10, 12] = [4, 2]. The "up" template is obtained by adding this distance to nominal:
[10, 12] + [4, 2] = [14, 14]
The "down" template should be symmetric, and have the same distance in the other direction of the nominal template:
[10, 12] - [4, 2] = [6, 10]
Both these templates still have a normalization effect, so they need to be normalized to match the integrated yield of the nominal template. The normalization factor is different for both samples, it is still (14+14) / (10+12) = 1.2727... in the "up" direction, but (6+10) / (10+12) = 0.7272... in the "down" direction. This leads to the following normalized templates:
“up”: [14,14] / 1.2727... = [11, 11]
“down”: [6, 10] / 0.7272... = [8.25, 13.75]
Both templates have the same total yield as the nominal one. The “down” template with this approach differs from the first method. The absolute distance of "up" and "down" from nominal is no longer the same.
Absolute versus relative distance
The examples above symmetrize by requiring that the templates satisfy up - nominal = nominal - down. The symmetrization could also require the relative distance to be the same, up / nominal = nominal / down. For the first method, with nominal template [10, 12] and "up" template [11, 11], the "down" template becomes [9.0909..., 13.0909...], which then would need to be normalized again to match the nominal yield, resulting in [9.016, 12.984].