Global Observable and Non-Gaussian constraint function

[StephanH]

Hi @xxiang4,

@xxiang4

When to define global observable? The example above set Gaussian mean m_mu_bkg as a global observable, but not the width s_mu_bkg . I’ve also seen other examples online that only define the mean in the constraint function as global observable. Why?

When you have a likelihood model that has a constraint term, then you need a “global observable”. A likelihood model that looks like this:

LH(data | parameters) = FitLikelihood(observable | paramA, paramB, paramLumi) * Gauss(lumi | paramLumi, sigmaLumi)

Has lumi as a “global observable”, that’s the observable that someone else measured for you. Think e.g. that the luminosity parameter should be close to 10 / fb, so you introduce the “observable” lumi, and you add a luminosity parameter, which you want to constrain to be close to 10. So you set the Gaussian to:

Gauss(10 | paramLumi, 1)

to get a 10% uncertainty for the luminosity parameter, centred around the external measurement of 10 that you got from the lumi group of your experiment.
Some people interchange the role of x and mu in the gaussian distribution, but that doesn’t matter because it’s symmetric. Sigma on the other hand is a parameter of the constraint term, it cannot be an observable. An observable is something that you can measure.

@xxiang4

Can I use other built-in function (ex. Crystal Ball) for the constrain function instead of Gaussian? If yes, do I need to define all four parameters of Crystal Ball as the global observable? Is it sufficient to set the constraint parameters to constants (ex. use setConstant(kTREU) ) instead of defining them as global observable?

Of course you can use a Crystall ball. Just do

LH_constr = LH_fit(x | param) * CrystalBall(globObs | param, ...)

Note that only the thing that has the role of x in the crystal ball is an observable. The parameters are parameters. And yes, most of the time you will set all parameters of the constraint term constant, because they should not be fitted. Only the parameter that is shared between the fit likelihood and the constraint likelihood will be floating in the fit.