Systematics in Hfitter

1. Systematics in Hfitter

2. Reminder: profiling nuisance parameters • Likelihood ratio is the most powerful discriminant between 2 hypotheses • What if the hypotheses depend on additional (“nuisance”) parameters ? • e.g. the background slope x. -> We “profile them away” : -> Wilks theorem: qm ~ c12 Note: 1) It’s a c2 2) Important here: it’s independent of q; profiling removes the nuisance parameters from the problem!

3. Systematics • Nuisance parameters come in 2 kinds • “Good”: parameters constrained by the fit. • The data tells us what their values are • e.g. background slope • “Bad” : not constrained • Need to introduce an outside constraint, which is added “by hand” to the likelihood. • These is what we call “Systematics” normally… • e.g. width of the signal peak sCB • Could let it float in the fit, but no sensitivity (until 2013 ?) • Can measure Z->ee, apply the result to gg : provides a constraint • Technically (on the sCBexample) • Can write sCB= sCB0(1 + a), a energy resolution • a is introduced since it is easily constrained : • it should be close to 0 (if sCB0computed with all corrections applied) • how far from 0 is a measure of the uncertainty (say 10% ?). • How to implement this precisely ?

4. Bayesian/Frequentist Hybrid treatment • 2 ways of dealing with the constraint in practice. First the Bayesian way, because it is more intuitive and more widespread. • Idea:assumea is distributed according to some PDF. • Obvious choice : a ~ Gaussian(0, 10%). • ais free, but the there is a penalty in the likelihood for a being different from 0. • Toys: • Each toy dataset must be thrown using a random value of a, drawn from the PDF. • Running over many toys effectively integrates out a. • Problem • a is a model parameter. Giving it a PDF is Bayesian • the PDF gives our “degree of belief” of where a should be.=> Not directly linked to something measured. (also, why a Gaussian ?)

5. Hfitter example (hfitter_Mgg_noCats_hybrid.dat) [Dependents] mgg = L(100 - 150) // gamma-gamma invariant mass [Models] component Signal = HggSigMggPdfBuilder() norm=(muSignal*nSignalSM; muSignal) component Background = HggBkgMggPdfBuilder() [Constraints] constraint dSig = RooGaussian("dSig", “mean_dSig", "sigma_dSig") [Parameters] nSignalSM = 1.225 C L(0 - 50000) muSignal = 1 L(-1000 - 10000) nBackground = 99 L(0 - 100000) [Signal] formula cbSigma = (cbSigma0*(1 + dSig)) dSig = 0 L(-1 - 10) mean_dSig = 0 C sigma_dSig = 0.10 C … The constraint on dSig : a Gaussian with specified parameters The cbSigma parameter is now given by a formula involving dSig dSig defined here. Allowed range is the important part.value really means starting value in the fit Constraint PDF parameters specified here (also could be in [Parameters])

6. “Frequentist” way • Idea: • like other nuisance parameters, a can be constrained in some way. • Problem here is that the constraint comes from another measurement • e.g. we could include a Z->ee sample in our model and fit everything simultaneously, getting a as a regular NP. But too complex.. • Solution: include the result from that other experiment • Use directly L(data’ |a) ? Too complex… • “Executive summary” : PDF(ames | a). e.g. ames ~ G(a, 10%). • Add this as a penalty term in the likelihood • Differences with Hybrid case • ames is a fixed measured value (if everything calibrated correctly, =0) • Note that a is now a PDF parameter. No PDF on a! (G gives a likelihood for a) • Similarities • ais still floating in the fit. Constraint still comes from penalty term • Note also that in this Gaussian case, L is the same as previously… but not always the case! • Toys: • There is a PDF on ames, so it should be randomized when generating toys • However, amesonly appears in the penalty term: all toys are in fact the same • ais just a parameter, it is not generated in the toys • Where does the smearing come in ? when fitting the toy, ais constrained by the value of ames.

7. Hfitter example (hfitter_Mgg_noCats_syst.dat) [Dependents] mgg = L(100 - 150) // gamma-gamma invariant mass [Models] component Signal = HggSigMggPdfBuilder() norm=(muSignal*nSignalSM; muSignal) component Background = HggBkgMggPdfBuilder() [Constraints] constraint dSig_aux = RooGaussian("dSig_aux", “dSig", "sigma_dSig") [Parameters] nSignalSM = 1.225 C L(0 - 50000) muSignal = 1 L(-1000 - 10000) nBackground = 99 L(0 - 100000) [Signal] formula cbSigma = (cbSigma0*(1 + dSig)) dSig = 0 L(-1 - 10) dSig_aux = 0 C sigma_dSig = 0.10 C … Constraint now on dSig_aux auxiliary measurement. dSig now a PDF parameter The cbSigma parameter is defined as previously dSig defined here, same as previously Now dSig_aux is constant (but can be randomized when generating toys).

8. Some results (2010 numbers w/smearing) Bayesian constraints on dSig, dEff, Gaussian with 10% width Frequentist constraints on dSig, dEff, Gaussian with 10% width

9. Some distributions

10. The way the constraint works • Bayesian case : • a0 = 0, but toys thrown with agen != 0=> in the fit, a drawn towards agen • Frequentist case : • ames randomized in toys=> in the fit, a drawn towards ames • Everything the same in this case, not true in distributions where a and a0 don’t play symmetric roles (e.g. Log-normal)

11. Results with 2011 numbers, Lognormal Bayesian constraints on dSig, dEff, Lognormal with 10% width Frequentist constraints on dSig, dEff, Lognormal with 10% width