Example of the BernsteinCorrection utility in RooStats.

The idea is that one has a distribution coming either from data or Monte Carlo (called "reality" in the macro) and a nominal model that is not sufficiently flexible to take into account the real distribution. One wants to take into account the systematic associated with this imperfect modeling by augmenting the nominal model with some correction term (in this case a polynomial). The BernsteinCorrection utility will import into your workspace a corrected model given by nominal(x) * poly_N(x), where poly_N is an n-th order polynomial in the Bernstein basis. The degree N of the polynomial is chosen by specifying the tolerance one has in adding an extra term to the polynomial. The Bernstein basis is nice because it only has positive-definite terms and works well with PDFs. Finally, the macro makes a plot of:

the data (drawn from 'reality'),
the best fit of the nominal model (blue)
and the best fit corrected model.

[#0] WARNING:InputArguments -- The parameter 'sigma' with range [0, 10] of the RooGaussian 'nominal' exceeds the safe range of (0, inf). Advise to limit its range.
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing dataset realityData
[#1] INFO:ObjectHandling -- RooWorkSpace::import(myWorksspace) changing name of dataset from  realityData to data
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::x
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooGaussian::nominal
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooConstVar::0
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::sigma
[#0] PROGRESS:Eval -- BernsteinCorrection::ImportCorrectedPdf -  Doing initial Fit with nominal model 
[#1] INFO:Fitting -- RooAbsPdf::fitTo(nominal_over_nominal_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- using generic CPU library compiled with no vectorizations
[#1] INFO:Fitting -- Creation of NLL object took 13.1477 ms
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_nominal_over_nominal_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 220.142 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 226.162 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 229.81 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 238.556 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 239.027 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 349.063 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_data) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooEffProd::corrected
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooBernstein::poly
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_0
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_1
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_2
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_3
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_4
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_5
[#1] INFO:ObjectHandling -- RooWorkspace::import(myWorksspace) importing RooRealVar::c_6
[#1] INFO:Eval -- ------ Begin Bernstein Correction Log --------
degree = 1 -log L(0) = 1216.78 -log L(1) = 1208.89 q = 15.7692 P(chi^2_1 > q) = 7.1557e-05
degree = 2 -log L(1) = 1208.89 -log L(2) = 1203.21 q = 11.3692 P(chi^2_1 > q) = 0.000746732
degree = 3 -log L(2) = 1203.21 -log L(3) = 1198.85 q = 8.72171 P(chi^2_1 > q) = 0.00314444
degree = 4 -log L(3) = 1198.85 -log L(4) = 1190.16 q = 17.3777 P(chi^2_1 > q) = 3.06393e-05
degree = 5 -log L(4) = 1190.16 -log L(5) = 1183.56 q = 13.1965 P(chi^2_1 > q) = 0.00028048
degree = 6 -log L(5) = 1183.56 -log L(6) = 1182.57 q = 1.98352 P(chi^2_1 > q) = 0.15902
------ End Bernstein Correction Log --------
[#1] INFO:Fitting -- RooAbsPdf::fitTo(nominal_over_nominal_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 164.909 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_nominal_over_nominal_Int[x]_realityData) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
Minuit2Minimizer: Minimize with max-calls 500 convergence for edm < 1 strategy 1
Minuit2Minimizer : Valid minimum - status = 0
FVAL  = 1216.77793416266263
Edm   = 4.16187387675607687e-07
Nfcn  = 19
sigma   = 1.18138  +/-  0.0315451   (limited)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(corrected_over_corrected_Int[x]) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- Creation of NLL object took 313.356 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_corrected_over_corrected_Int[x]_realityData) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- [fitFCN] No discrete parameters, performing continuous minimization only
Minuit2Minimizer: Minimize with max-calls 3500 convergence for edm < 1 strategy 1
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
Minuit2Minimizer : Valid minimum - status = 0
FVAL  = 1182.56856204371252
Edm   = 0.00013228457688265106
Nfcn  = 185
c_1     = 3.18344  +/-  0.786592 (limited)
c_2     = 4.74802e-06    +/-  3.10471  (limited)
c_3     = 1.13228e-06    +/-  1.51799  (limited)
c_4     = 0.966537    +/-  2.16438  (limited)
c_5     = 0.216445    +/-  67.7996  (limited)
c_6     = 10.4408  +/-  20.6953  (limited)
sigma   = 1.26669  +/-  0.210811 (limited)
[#1] INFO:NumericIntegration -- RooRealIntegral::init(corrected_Int[x]) using numeric integrator RooIntegrator1D to calculate Int(x)
Correction based on Bernstein Poly of degree  6

 
import sys
import ROOT
 
# set range of observable
lowRange = -1
highRange = 5
 
# make a RooRealVar for the observable
x = ROOT.RooRealVar("x", "x", lowRange, highRange)
 
# true model
narrow = ROOT.RooGaussian("narrow", "", x, ROOT.RooFit.RooConst(0.0), ROOT.RooFit.RooConst(0.8))
wide = ROOT.RooGaussian("wide", "", x, ROOT.RooFit.RooConst(0.0), ROOT.RooFit.RooConst(2.0))
reality = ROOT.RooAddPdf("reality", "", [narrow, wide], ROOT.RooFit.RooConst(0.8))
 
data = reality.generate(x, 1000)
 
# nominal model
sigma = ROOT.RooRealVar("sigma", "", 1.0, 0, 10)
nominal = ROOT.RooGaussian("nominal", "", x, ROOT.RooFit.RooConst(0.0), sigma)
 
wks = ROOT.RooWorkspace("myWorksspace")
 
wks.Import(data, Rename="data")
wks.Import(nominal)
 
if ROOT.TClass.GetClass("ROOT::Minuit2::Minuit2Minimizer"):
    # use Minuit2 if ROOT was built with support for it:
    ROOT.Math.MinimizerOptions.SetDefaultMinimizer("Minuit2")
 
# The tolerance sets the probability to add an unnecessary term.
# lower tolerance will add fewer terms, while higher tolerance
# will add more terms and provide a more flexible function.
tolerance = 0.05
bernsteinCorrection = ROOT.RooStats.BernsteinCorrection(tolerance)
degree = bernsteinCorrection.ImportCorrectedPdf(wks, "nominal", "x", "data")
 
if degree < 0:
    ROOT.Error("rs_bernsteinCorrection", "Bernstein correction failed !")
    sys.exit()
 
print("Correction based on Bernstein Poly of degree ", degree)
 
frame = x.frame()
data.plotOn(frame)
# plot the best fit nominal model in blue
nominal.fitTo(data, PrintLevel=0)
nominal.plotOn(frame)
 
# plot the best fit corrected model in red
corrected = wks["corrected"]
if not corrected:
    sys.exit()
 
# fit corrected model
corrected.fitTo(data, PrintLevel=0)
corrected.plotOn(frame, LineColor="r")
 
# plot the correction term (* norm constant) in dashed green
# should make norm constant just be 1, not depend on binning of data
poly = wks["poly"]
if poly:
    poly.plotOn(frame, LineColor="g", LineStyle="--")
 
# this is a switch to check the sampling distribution
# of -2 log LR for two comparisons:
# the first is for n-1 vs. n degree polynomial corrections
# the second is for n vs. n+1 degree polynomial corrections
# Here we choose n to be the one chosen by the tolerance
# criterion above, eg. n = "degree" in the code.
# Setting this to true is takes about 10 min.
checkSamplingDist = True
numToyMC = 20  # increase this value for sensible results
 
c1 = ROOT.TCanvas()
if checkSamplingDist:
    c1.Divide(1, 2)
    c1.cd(1)
 
frame.Draw()
ROOT.gPad.Update()
 
if checkSamplingDist:
    # check sampling dist
    ROOT.Math.MinimizerOptions.SetDefaultPrintLevel(-1)
    samplingDist = ROOT.TH1F("samplingDist", "", 20, 0, 10)
    samplingDistExtra = ROOT.TH1F("samplingDistExtra", "", 20, 0, 10)
    bernsteinCorrection.CreateQSamplingDist(
        wks, "nominal", "x", "data", samplingDist, samplingDistExtra, degree, numToyMC
    )
 
    c1.cd(2)
    samplingDistExtra.SetLineColor("kRed")
    samplingDistExtra.Draw()
    samplingDist.Draw("same")
 
c1.SaveAs("rs_bernsteinCorrection.png")

Date: June 2022

Authors: Artem Busorgin, Kyle Cranmer (C++ version)

Definition in file rs_bernsteinCorrection.py.