Obtaining the likelihood function after fitting

[StephanH]

@Lepton86

I will try to find something to demonstrate what I mean. But for now all I want to do is smear the resultant profile likelihood in sigYield, not constrain it (even if it is not mathematically well defined). That’s what I’m trying to figure out how to do in RooFit.

I see. The reason there’s no obvious way to do it is probably that it’s not well defined.

@Lepton86

It is not explicitly a convolution, it’s just a way to introduce the systematics to make the profile wider. The high error should then naively be approximately sqrt(stat. err^2 + syst. err^2), where the stat err is the error from the normal fit. That’s what I’m ultimately trying to get to.

If stat error and systematic error are sufficiently uncorrelated, that’s what you will get when you introduce the systematic in the “standard way”.

@Lepton86

Yes. Did I make that mistake somewhere?

Yes, here:

@Lepton86

syst_nll = TF1("syst_nll","-TMath::Log(TMath::Gaus(x,[0],[1],0))", sigLO, sigHI)

@Lepton86

I’ve tried that. But that constrains (narrows) the profile of the POI, unless I misunderstood something.

No, no misunderstanding. It is supposed to narrow it. Let me reiterate:

• Introduce a nuisance parameter → Profile widens (but maybe more than you want, because the effect is unconstrained.
• Introduce a nuisance parameter + constrain it → Profile widens, but by the “correct” amount.
• Introduce a nuisance parameter + constrain it infinitely strongly → Profile remains the same as if you didn’t introduce the NP.

@Lepton86

The nuisance parameter ( sigScaler in your solution) can vary such that the profile is not at all parabolic and the high MINOS error is much larger than the approximation sqrt(stat. err^2 + syst. error^2). For example.

Corresponding to the fit result

But that’s exactly right! If you introduce the nuisance parameter, and don’t constrain it, your errors will explode! You basically say: The signal systematic can be as strong as the fitter desires, even infinitely strong. Here, you completed step 1 so to say.

@Lepton86

If I set the lower bound on sigScalar to be higher than zero, e.g. 0.5, I get a more parabolic profile and the error is closer to what I expect, but MINOS gives the “at limit” warning.

The range of the parameters should not be used as constraints. They are really hard limits, which convey physical limits (e.g. that a signal scaler cannot be negative). What’s missing now is that you say:
“The signal scaler is allowed to vary between 0.8 and 1.2” (or whatever is appropriate for your systematic). This is what you do by additionally introducing a constraint on the nuisance parameter.

Again:

• Adding a NP increases errors. You did that.
• Constraining the NP will get it to the “right amount”.