# Simultaneous fit and upper limit with constraints

Dear Experts,

I am performing a simultaneous fit in 3 fit categories, to determine the upper limit on two signal branching fractions. Fit categories 1 and 2 contain a background component, whose Yield is expressed as follows:

N_BKG(cat) = Expexted_BKG(cat)/Expexted_BKG_total *N_BKG_total,

where N_BKG_total is a shared parameter between Fit categories 1 and 2.
Furthermore, N_BKG_total is a Gauss constraint.

For the limit calculation I use the AsymptoticCalculator, and to take the Gauss constraint into account I basically followed the discussion from this post:

root-forum.cern.ch/t/upper-limit-calculation/34559

However, since the constrained component is not present in one of the Fit categories, I am unsure if my implementation makes sense:

``````#Signal Yields
ws.factory(f'expr::N_sig1{label_fitcat}("(BF_sig1/BF_Normalisation)*Normalisation_Yield*(eps_sig1{label_fitcat}/eps_Normalisation)",BF_sig1,BF_Normalisation,  Normalisation_Yield,eps_sig1{label_fitcat},eps_Normalisation)')
ws.factory(f'expr::N_sig2{label_fitcat}("(BF_sig2/BF_Normalisation)*Normalisation_Yield*(eps_sig2{label_fitcat}/eps_Normalisation)*(fs_fu)",BF_sig2,BF_Normalisation, Normalisation_Yield,eps_sig2{label_fitcat},eps_Normalisation,fs_fu)')

#Phys BKG Yield
if fit_category_name =="Fit_category_1" or fit_category_name =="Fit_category_2":
Total_expexted_Bkg = 2.4
Expexted_Yield_Bkg= fit_category_dicc["DB_parameters"]["Expexted_Bkg"]
WS(ws,r.RooRealVar(f"Scale_Bkg_{label_fitcat}",  f"Scale_Bkg_{label_fitcat}",Expexted_Yield_Bkg/Total_expexted_Bkg))
ws.factory(f'expr::N_Bkg_{label_fitcat}("(Scale_Bkg_{label_fitcat})*N_Bkg_total",Scale_Bkg_{label_fitcat},N_Bkg_total)')

#####################################################################
#Constrains on BKG Yield
ws.factory(f"Gaussian::Constrain_BKG(Bkg_global[2.4,0,30],N_Bkg_total ,sigma_const[1.7])")
ws.var("Bkg_global").setConstant(True)
ws.defineSet("Global_Obs","Bkg_global")
ws.defineSet("NuisanceParameters","Ncomb_Fitcat1,Ncomb_Fitcat2,Ncomb_Fitcat3,exp_coef_Fitcat1,exp_coef_Fitcat2,exp_coef_Fitcat3")
# #Is this right?
ws.factory(f"PROD::PDF_Bkg_Fitcat1const(PDF_Bkg_Fitcat1,Constrain_BKG)")
ws.factory(f"PROD::PDF_Bkg_Fitcat2const(PDF_Bkg_Fitcat2,Constrain_BKG)")

#####################################################################
#Build Model
#Fitcat 1 and 2 contain the BKG:
ws.factory(f"SUM::Model_Fitcat1(N_sig1Fitcat1*PDF_sig1_Fitcat1,"
f"N_sig2Fitcat1*PDF_sig2_Fitcat1,"
f"Ncomb_Fitcat1*PDF_comb_bkg_Fitcat1,"
f"N_Bkg_Fitcat1*PDF_Bkg_Fitcat1const)")
ws.factory(f"SUM::Model_Fitcat2(N_sig1Fitcat2*PDF_sig1_Fitcat2,"
f"N_sig2Fitcat2*PDF_sig2_Fitcat2,"
f"Ncomb_Fitcat2*PDF_comb_bkg_Fitcat2,"
f"N_Bkg_Fitcat2*PDF_Bkg_Fitcat2const)")
# Fitcat3 does not contain the BKG
ws.factory(f"SUM::Model_Fitcat3(N_sig1Fitcat3*PDF_sig1_Fitcat3,"
f"N_sig2Fitcat3*PDF_sig2_Fitcat3,"
f"Ncomb_Fitcat3*PDF_comb_bkg_Fitcat3)")

#####################################################################
# Build  RooSimultanous
roo_category = WS(ws,r.RooCategory("category","category"))
roo_category.defineType("Fitcat1")
roo_category.defineType("Fitcat2")
roo_category.defineType("Fitcat3")
dsmap = r.std.map('string, RooDataSet*')()
dsmap["Fitcat1"] = data_set_holder["Fitcat1"]
dsmap["Fitcat2"] = data_set_holder["Fitcat2"]
dsmap["Fitcat3"] = data_set_holder["Fitcat3"]
data_incat = WS(ws,r.RooDataSet("data","data",r.RooArgSet(mass),r.RooFit.Index(roo_category),r.RooFit.Import(dsmap),))
model = WS(ws,r.RooSimultaneous("Model_Final", "Model_Final", roo_category))

#####################################################################
# Fit
_fitResults= model.fitTo(data_incat, args_fit)
``````

I am especially insecure if I need to multiply PDF of the BKG with the constraint or the full Model and how this would look in terms of the Likelihood.

Furhtermore, I implemented the upper limit calculation for one of the two signal componets:

``````#####################################################################
# Calculate Limit for sig1  only
#####################################################################

#Set sig2 to zero
ws.var("BF_sig2").setVal(0)
ws.var("BF_sig2").setConstant(r.kTRUE)

#signal + background mode
sb_model = r.RooStats.ModelConfig("S+B_model",ws)
sb_model.SetPdf(ws.pdf("Model_Final"))
sb_model.SetGlobalObservables(ws.set("Global_Obs"))
sb_model.SetNuisanceParameters(ws.set("NuisanceParameters"))
sb_model.SetParametersOfInterest("BF_sig1")
sb_model.SetObservables(r.RooArgSet(ws.var("B_DTF_M_GeV"), ws.cat("category")))
sb_model.SetSnapshot(r.RooArgSet(ws.var("BF_sig1")))
val_sig1 = ws.var("BF_sig1").getVal()
# bkg ony model
b_model = r.RooStats.ModelConfig("B_model",ws)
b_model.SetPdf(ws.pdf("Model_Final"))
b_model.SetGlobalObservables(ws.set("Global_Obs"))
b_model.SetNuisanceParameters(ws.set("NuisanceParameters"))
ws.var("BF_sig1").setVal(0)
ws.var("BF_sig1").setConstant(r.kTRUE)
b_model.SetParametersOfInterest("BF_sig1")
b_model.SetObservables(r.RooArgSet(ws.var("B_DTF_M_GeV"), ws.cat("category")))
b_model.SetSnapshot(r.RooArgSet(ws.var("BF_sig1")))
ws.var("BF_sig1").setVal(val_sig1)
ws.var("BF_sig1").setConstant(r.kFALSE)
ac = r.RooStats.AsymptoticCalculator(ws.data("data"), b_model, sb_model)
ac.SetOneSided(True)
calc = r.RooStats.HypoTestInverter(ac)
calc.UseCLs(True)
calc.SetConfidenceLevel(0.95)
poimin = 0
poimax =4e-9
npoints=20
calc.SetFixedScan(npoints,poimin,poimax)
results = calc.GetInterval()
upperLimit = results.UpperLimit()
ulError = results.UpperLimitEstimatedError()
expectedLimit = results.GetExpectedUpperLimit(0)
``````

So from what I understood is that for the limit calculation, it is essential to set the Global Observables
and the Nuisance Parameters, to take the constraint into account. Do I also have to include the N_BKG_total in the Nuisance Parameters? And how do I deal with the second signal BF? I assume I have to set it to zero while I am determining the limit for the first BF?

I would appreciate any advice regarding this points.

Thank you,

Seb

1 Like

Welcome to the ROOT forum
I think @moneta can help.

2 Likes

Hi, me and @sebmnt are working together on this and we got confused on how to properly tell the Asymptotic calculator to apply gauss constraint in some of the categories of the fit.
We are quite confident if we would not rely on the factory constructors and roosimultanous PDF with indexed categories on how to do It juggling the likelihoods (plain sums of the log likelihood of each category and a single term for the constrained parameter), but once the factory Is used and we introduce gauss constraint It becomes unclear if the code would add for each category the same likelihood penalty if the constrained parameter Is shared among some of the categories or It would do It Only once as It should.
If there Is anyone which can provide a feedback on the original post would be great
. Thanks again
Renato