I would make this topic as rich and complete as possible.

Some asymptotic formula: https://cds.cern.ch/record/2643488/files/ATL-COM-GEN-2018-026.pdf

Auxiliary measurements provide uncertainties for measured variables. Those variables are used in analysis. Explicitly variables are like luminosity, calibration factor and scale factor inside a certain object bin of pt vs eta; implicitly, they can be "the effect of changing your smearing algorithm", "changing your underlying parton distribution function", etc. They are given by auxiliary measurement in a 1 sigma deviation manner.

The 1 sigma variation will be usually assigned to a gaussian constraint, , and prediction will be sth like (It's the simplest linear case). This is based on the assumption that 1 sigma variation on your parameter will have a 1 sigma variation impact on the final result.

See here for some introduction to systematic uncertainty. It also introduce something about profile likelihood, which absorb the nuisance parameters

[ Pekka K. Definition and Treatement of Systematic Uncertainties in High Energy Physics and Astrophysics].

A fit is a procedure to find minimum/maximum of a certain metric, to see the ability of your model to describe the data.

Errors are usually calculated assuming 1 sigma deviation from nominal, by convention. So nuisance parameter are usually assigned to a Gaussian constraint with sigma=1. If the estimated error given by fitting algorithms is less than 1, we have overconstrain. If it's over 1, we have underconstrain.

The estimated errors on the nuisance parameter given by the the fitting algorithm are what the algorithm defines 1sigma error. So if you have overconstrain, for example 0.9, the algorithm thought your error should be 0.9 * variation_1sigma. In this case, you might have overestimated your error, or you can say it's too conservative. But also, it could be due to your response model is too simple.

See more here: [W. Verkerke *Practical Statistics - Part III* ]

Error of a POI/fitting parameter are given by:

Migrad: quick esetimate.

Hesse: square of second derivative at best fit point. This is assuming parabola shape NLL.

Minos: find intersection of min_NLL + 0.5 and profile scan of POI.

Minuit2:

- Likelihood : multiplication of simple probability in each measurement(usually means yield in bin), as an combined probability of observing this kind of data.
- : negative log likelihood . It has some nice property in fitting.
- : It's simply when is gaussian-like, and is used in simple fitting.

- Will Buttinger's 'Learning Roostats': a quite complete hand-by-hand example for constructing workspace in old & new ways.
- Systematic uncertainties and profiling: some slides,
*unread.* - Statistics Tools in ATLAS.

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