HybridInstructional.C

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/// \file
/// \ingroup tutorial_roostats
/// \notebook -js
/// Example demonstrating usage of HybridCalcultor
///
/// A hypothesis testing example based on number counting
/// with background uncertainty.
///
/// NOTE: This example must be run with the ACLIC (the + option ) due to the
/// new class that is defined.
///
/// This example:
///  - demonstrates the usage of the HybridCalcultor (Part 4-6)
///  - demonstrates the numerical integration of RooFit (Part 2)
///  - validates the RooStats against an example with a known analytic answer
///  - demonstrates usage of different test statistics
///  - explains subtle choices in the prior used for hybrid methods
///  - demonstrates usage of different priors for the nuisance parameters
///  - demonstrates usage of PROOF
///
/// The basic setup here is that a main measurement has observed x events with an
/// expectation of s+b.  One can choose an ad hoc prior for the uncertainty on b,
/// or try to base it on an auxiliary measurement.  In this case, the auxiliary
/// measurement (aka control measurement, sideband) is another counting experiment
/// with measurement y and expectation tau*b.  With an 'original prior' on b,
/// called \f$\eta(b)\f$ then one can obtain a posterior from the auxiliary measurement
/// \f$\pi(b) = \eta(b) * Pois(y|tau*b).\f$  This is a principled choice for a prior
/// on b in the main measurement of x, which can then be treated in a hybrid
/// Bayesian/Frequentist way.  Additionally, one can try to treat the two
/// measurements simultaneously, which is detailed in Part 6 of the tutorial.
///
/// This tutorial is related to the FourBin.C tutorial in the modeling, but
/// focuses on hypothesis testing instead of interval estimation.
///
/// More background on this 'prototype problem' can be found in the
/// following papers:
///
///  - Evaluation of three methods for calculating statistical significance
///    when incorporating a systematic uncertainty into a test of the
///    background-only hypothesis for a Poisson process
///    Authors: Robert D. Cousins, James T. Linnemann, Jordan Tucker
///    http://arxiv.org/abs/physics/0702156
///    NIM  A 595 (2008) 480--501
///
///  - Statistical Challenges for Searches for New Physics at the LHC
///    Author: Kyle Cranmer
///    http://arxiv.org/abs/physics/0511028
///
///  - Measures of Significance in HEP and Astrophysics
///    Authors J. T. Linnemann
///    http://arxiv.org/abs/physics/0312059
///
/// \macro_image
/// \macro_output
/// \macro_code
///
/// \authors Kyle Cranmer, Wouter Verkerke, and Sven Kreiss
 
#include "RooGlobalFunc.h"
#include "RooRealVar.h"
#include "RooProdPdf.h"
#include "RooWorkspace.h"
#include "RooDataSet.h"
#include "TCanvas.h"
#include "TStopwatch.h"
#include "TH1.h"
#include "RooPlot.h"
#include "RooMsgService.h"
 
#include "RooStats/NumberCountingUtils.h"
 
#include "RooStats/HybridCalculator.h"
#include "RooStats/ToyMCSampler.h"
#include "RooStats/HypoTestPlot.h"
 
#include "RooStats/ProfileLikelihoodTestStat.h"
#include "RooStats/SimpleLikelihoodRatioTestStat.h"
#include "RooStats/RatioOfProfiledLikelihoodsTestStat.h"
#include "RooStats/MaxLikelihoodEstimateTestStat.h"
 
using namespace RooFit;
using namespace RooStats;
 
// ----------------------------------
// A New Test Statistic Class for this example.
// It simply returns the sum of the values in a particular
// column of a dataset.
// You can ignore this class and focus on the macro below
class BinCountTestStat : public TestStatistic {
public:
   BinCountTestStat(void) : fColumnName("tmp") {}
   BinCountTestStat(string columnName) : fColumnName(columnName) {}
 
   virtual Double_t Evaluate(RooAbsData &data, RooArgSet & /*nullPOI*/)
   {
      // This is the main method in the interface
      Double_t value = 0.0;
      for (int i = 0; i < data.numEntries(); i++) {
         value += data.get(i)->getRealValue(fColumnName.c_str());
      }
      return value;
   }
   virtual const TString GetVarName() const { return fColumnName; }
 
private:
   string fColumnName;
 
protected:
   ClassDef(BinCountTestStat, 1)
};
 
ClassImp(BinCountTestStat)
 
// ----------------------------------
// The Actual Tutorial Macro
 
void HybridInstructional(int ntoys = 6000)
{
   double nToysRatio = 20;      // ratio Ntoys Null/ntoys ALT
 
   // This tutorial has 6 parts
   // Table of Contents
   // Setup
   //   1. Make the model for the 'prototype problem'
   // Special cases
   //   2. Use RooFit's direct integration to get p-value & significance
   //   3. Use RooStats analytic solution for this problem
   // RooStats HybridCalculator -- can be generalized
   //   4. RooStats ToyMC version of 2. & 3.
   //   5. RooStats ToyMC with an equivalent test statistic
   //   6. RooStats ToyMC with simultaneous control & main measurement
 
   // It takes ~4 min without PROOF and ~2 min with PROOF on 4 cores.
   // Of course, everything looks nicer with more toys, which takes longer.
 
   TStopwatch t;
   t.Start();
   TCanvas *c = new TCanvas;
   c->Divide(2, 2);
 
   // ----------------------------------------------------
   // P A R T   1  :  D I R E C T   I N T E G R A T I O N
   // ====================================================
   // Make model for prototype on/off problem
   // $Pois(x | s+b) * Pois(y | tau b )$
   // for Z_Gamma, use uniform prior on b.
   RooWorkspace *w = new RooWorkspace("w");
   w->factory("Poisson::px(x[150,0,500],sum::splusb(s[0,0,100],b[100,0.1,300]))");
   w->factory("Poisson::py(y[100,0.1,500],prod::taub(tau[1.],b))");
   w->factory("PROD::model(px,py)");
   w->factory("Uniform::prior_b(b)");
 
   // We will control the output level in a few places to avoid
   // verbose progress messages.  We start by keeping track
   // of the current threshold on messages.
   RooFit::MsgLevel msglevel = RooMsgService::instance().globalKillBelow();
 
   // Use PROOF-lite on multi-core machines
   ProofConfig *pc = NULL;
   // uncomment below if you want to use PROOF
   // ~~~
   // pc = new ProofConfig(*w, 4, "workers=4", kFALSE); // machine with 4 cores
   // pc = new ProofConfig(*w, 2, "workers=2", kFALSE); // machine with 2 cores
   // ~~~
 
   // ----------------------------------------------------
   // P A R T   2  :  D I R E C T   I N T E G R A T I O N
   // ====================================================
   // This is not the 'RooStats' way, but in this case the distribution
   // of the test statistic is simply x and can be calculated directly
   // from the PDF using RooFit's built-in integration.
   // Note, this does not generalize to situations in which the test statistic
   // depends on many events (rows in a dataset).
 
   // construct the Bayesian-averaged model (eg. a projection pdf)
   // $p'(x|s) = \int db p(x|s+b) * [ p(y|b) * prior(b) ]$
   w->factory("PROJ::averagedModel(PROD::foo(px|b,py,prior_b),b)");
 
   RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR); // lower message level
   // plot it, red is averaged model, green is b known exactly, blue is s+b av model
   RooPlot *frame = w->var("x")->frame(Range(50, 230));
   w->pdf("averagedModel")->plotOn(frame, LineColor(kRed));
   w->pdf("px")->plotOn(frame, LineColor(kGreen));
   w->var("s")->setVal(50.);
   w->pdf("averagedModel")->plotOn(frame, LineColor(kBlue));
   c->cd(1);
   frame->Draw();
   w->var("s")->setVal(0.);
 
   // compare analytic calculation of Z_Bi
   // with the numerical RooFit implementation of Z_Gamma
   // for an example with x = 150, y = 100
 
   // numeric RooFit Z_Gamma
   w->var("y")->setVal(100);
   w->var("x")->setVal(150);
   std::unique_ptr<RooAbsReal> cdf{w->pdf("averagedModel")->createCdf(*w->var("x"))};
   cdf->getVal(); // get ugly print messages out of the way
   cout << "-----------------------------------------" << endl;
   cout << "Part 2" << endl;
   cout << "Hybrid p-value from direct integration = " << 1 - cdf->getVal() << endl;
   cout << "Z_Gamma Significance  = " << PValueToSignificance(1 - cdf->getVal()) << endl;
   RooMsgService::instance().setGlobalKillBelow(msglevel); // set it back
 
   // ---------------------------------------------
   // P A R T   3  :  A N A L Y T I C   R E S U L T
   // =============================================
   // In this special case, the integrals are known analytically
   // and they are implemented in RooStats::NumberCountingUtils
 
   // analytic Z_Bi
   double p_Bi = NumberCountingUtils::BinomialWithTauObsP(150, 100, 1);
   double Z_Bi = NumberCountingUtils::BinomialWithTauObsZ(150, 100, 1);
   cout << "-----------------------------------------" << endl;
   cout << "Part 3" << endl;
   std::cout << "Z_Bi p-value (analytic): " << p_Bi << std::endl;
   std::cout << "Z_Bi significance (analytic): " << Z_Bi << std::endl;
   t.Stop();
   t.Print();
   t.Reset();
   t.Start();
 
   // -------------------------------------------------------------
   // P A R T   4  :  U S I N G   H Y B R I D   C A L C U L A T O R
   // =============================================================
   // Now we demonstrate the RooStats HybridCalculator.
   //
   // Like all RooStats calculators it needs the data and a ModelConfig
   // for the relevant hypotheses.  Since we are doing hypothesis testing
   // we need a ModelConfig for the null (background only) and the alternate
   // (signal+background) hypotheses.  We also need to specify the PDF,
   // the parameters of interest, and the observables.  Furthermore, since
   // the parameter of interest is floating, we need to specify which values
   // of the parameter corresponds to the null and alternate (eg. s=0 and s=50)
   //
   // define some sets of variables obs={x} and poi={s}
   // note here, x is the only observable in the main measurement
   // and y is treated as a separate measurement, which is used
   // to produce the prior that will be used in this calculation
   // to randomize the nuisance parameters.
   w->defineSet("obs", "x");
   w->defineSet("poi", "s");
 
   // create a toy dataset with the x=150
   RooDataSet *data = new RooDataSet("d", "d", *w->set("obs"));
   data->add(*w->set("obs"));
 
   // Part 3a : Setup ModelConfigs
   // -------------------------------------------------------
   // create the null (background-only) ModelConfig with s=0
   ModelConfig b_model("B_model", w);
   b_model.SetPdf(*w->pdf("px"));
   b_model.SetObservables(*w->set("obs"));
   b_model.SetParametersOfInterest(*w->set("poi"));
   w->var("s")->setVal(0.0); // important!
   b_model.SetSnapshot(*w->set("poi"));
 
   // create the alternate (signal+background) ModelConfig with s=50
   ModelConfig sb_model("S+B_model", w);
   sb_model.SetPdf(*w->pdf("px"));
   sb_model.SetObservables(*w->set("obs"));
   sb_model.SetParametersOfInterest(*w->set("poi"));
   w->var("s")->setVal(50.0); // important!
   sb_model.SetSnapshot(*w->set("poi"));
 
   // Part 3b : Choose Test Statistic
   // ----------------------------------
   // To make an equivalent calculation we need to use x as the test
   // statistic.  This is not a built-in test statistic in RooStats
   // so we define it above.  The new class inherits from the
   // RooStats::TestStatistic interface, and simply returns the value
   // of x in the dataset.
 
   BinCountTestStat binCount("x");
 
   // Part 3c : Define Prior used to randomize nuisance parameters
   // -------------------------------------------------------------
   //
   // The prior used for the hybrid calculator is the posterior
   // from the auxiliary measurement y.  The model for the aux.
   // measurement is Pois(y|tau*b), thus the likelihood function
   // is proportional to (has the form of) a Gamma distribution.
   // if the 'original prior' $\eta(b)$ is uniform, then from
   // Bayes's theorem we have the posterior:
   // $\pi(b) = Pois(y|tau*b) * \eta(b)$
   // If $\eta(b)$ is flat, then we arrive at a Gamma distribution.
   // Since RooFit will normalize the PDF we can actually supply
   // $py=Pois(y,tau*b)$ that will be equivalent to multiplying by a uniform.
   //
   // Alternatively, we could explicitly use a gamma distribution:
   // `w->factory("Gamma::gamma(b,sum::temp(y,1),1,0)");`
   //
   // or we can use some other ad hoc prior that do not naturally
   // follow from the known form of the auxiliary measurement.
   // The common choice is the equivalent Gaussian:
   w->factory("Gaussian::gauss_prior(b,y, expr::sqrty('sqrt(y)',y))");
   // this corresponds to the "Z_N" calculation.
   //
   // or one could use the analogous log-normal prior
   w->factory("Lognormal::lognorm_prior(b,y, expr::kappa('1+1./sqrt(y)',y))");
   //
   // Ideally, the HybridCalculator would be able to inspect the full
   // model $Pois(x | s+b) * Pois(y | tau b )$ and be given the original
   // prior $\eta(b)$ to form $\pi(b) = Pois(y|tau*b) * \eta(b)$.
   // This is not yet implemented because in the general case
   // it is not easy to identify the terms in the PDF that correspond
   // to the auxiliary measurement.  So for now, it must be set
   // explicitly with:
   //  - ForcePriorNuisanceNull()
   //  - ForcePriorNuisanceAlt()
   // the name "ForcePriorNuisance" was chosen because we anticipate
   // this to be auto-detected, but will leave the option open
   // to force to a different prior for the nuisance parameters.
 
   // Part 3d : Construct and configure the HybridCalculator
   // -------------------------------------------------------
 
   HybridCalculator hc1(*data, sb_model, b_model);
   ToyMCSampler *toymcs1 = (ToyMCSampler *)hc1.GetTestStatSampler();
   toymcs1->SetNEventsPerToy(1);         // because the model is in number counting form
   toymcs1->SetTestStatistic(&binCount); // set the test statistic
   hc1.SetToys(ntoys, ntoys / nToysRatio);
   hc1.ForcePriorNuisanceAlt(*w->pdf("py"));
   hc1.ForcePriorNuisanceNull(*w->pdf("py"));
   // if you wanted to use the ad hoc Gaussian prior instead
   // ~~~
   //  hc1.ForcePriorNuisanceAlt(*w->pdf("gauss_prior"));
   //  hc1.ForcePriorNuisanceNull(*w->pdf("gauss_prior"));
   // ~~~
   // if you wanted to use the ad hoc log-normal prior instead
   // ~~~
   //  hc1.ForcePriorNuisanceAlt(*w->pdf("lognorm_prior"));
   //  hc1.ForcePriorNuisanceNull(*w->pdf("lognorm_prior"));
   // ~~~
 
   // enable proof
   // NOTE: This test statistic is defined in this macro, and is not
   // working with PROOF currently.  Luckily test stat is fast to evaluate.
   //  `if(pc) toymcs1->SetProofConfig(pc);`
 
   // these lines save current msg level and then kill any messages below ERROR
   RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR);
   // Get the result
   HypoTestResult *r1 = hc1.GetHypoTest();
   RooMsgService::instance().setGlobalKillBelow(msglevel); // set it back
   cout << "-----------------------------------------" << endl;
   cout << "Part 4" << endl;
   r1->Print();
   t.Stop();
   t.Print();
   t.Reset();
   t.Start();
 
   c->cd(2);
   HypoTestPlot *p1 = new HypoTestPlot(*r1, 30); // 30 bins, TS is discrete
   p1->Draw();
 
   // -------------------------------------------------------------------------
   // # P A R T   5  :  U S I N G   H Y B R I D   C A L C U L A T O R
   // # W I T H   A N   A L T E R N A T I V E   T E S T   S T A T I S T I C
   //
   // A likelihood ratio test statistics should be 1-to-1 with the count x
   // when the value of b is fixed in the likelihood.  This is implemented
   // by the SimpleLikelihoodRatioTestStat
 
   SimpleLikelihoodRatioTestStat slrts(*b_model.GetPdf(), *sb_model.GetPdf());
   slrts.SetNullParameters(*b_model.GetSnapshot());
   slrts.SetAltParameters(*sb_model.GetSnapshot());
 
   // HYBRID CALCULATOR
   HybridCalculator hc2(*data, sb_model, b_model);
   ToyMCSampler *toymcs2 = (ToyMCSampler *)hc2.GetTestStatSampler();
   toymcs2->SetNEventsPerToy(1);
   toymcs2->SetTestStatistic(&slrts);
   hc2.SetToys(ntoys, ntoys / nToysRatio);
   hc2.ForcePriorNuisanceAlt(*w->pdf("py"));
   hc2.ForcePriorNuisanceNull(*w->pdf("py"));
   // if you wanted to use the ad hoc Gaussian prior instead
   // ~~~
   //  hc2.ForcePriorNuisanceAlt(*w->pdf("gauss_prior"));
   //  hc2.ForcePriorNuisanceNull(*w->pdf("gauss_prior"));
   // ~~~
   // if you wanted to use the ad hoc log-normal prior instead
   // ~~~
   //  hc2.ForcePriorNuisanceAlt(*w->pdf("lognorm_prior"));
   //  hc2.ForcePriorNuisanceNull(*w->pdf("lognorm_prior"));
   // ~~~
 
   // enable proof
   if (pc)
      toymcs2->SetProofConfig(pc);
 
   // these lines save current msg level and then kill any messages below ERROR
   RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR);
   // Get the result
   HypoTestResult *r2 = hc2.GetHypoTest();
   cout << "-----------------------------------------" << endl;
   cout << "Part 5" << endl;
   r2->Print();
   t.Stop();
   t.Print();
   t.Reset();
   t.Start();
   RooMsgService::instance().setGlobalKillBelow(msglevel);
 
   c->cd(3);
   HypoTestPlot *p2 = new HypoTestPlot(*r2, 30); // 30 bins
   p2->Draw();
 
   // -----------------------------------------------------------------------------
   // # P A R T   6  :  U S I N G   H Y B R I D   C A L C U L A T O R   W I T H   A N   A L T E R N A T I V E   T E S T
   // # S T A T I S T I C   A N D   S I M U L T A N E O U S   M O D E L
   //
   // If one wants to use a test statistic in which the nuisance parameters
   // are profiled (in one way or another), then the PDF must constrain b.
   // Otherwise any observation x can always be explained with s=0 and b=x/tau.
   //
   // In this case, one is really thinking about the problem in a
   // different way.  They are considering x,y simultaneously.
   // and the PDF should be $Pois(x | s+b) * Pois(y | tau b )$
   // and the set 'obs' should be {x,y}.
 
   w->defineSet("obsXY", "x,y");
 
   // create a toy dataset with the x=150, y=100
   w->var("x")->setVal(150.);
   w->var("y")->setVal(100.);
   RooDataSet *dataXY = new RooDataSet("dXY", "dXY", *w->set("obsXY"));
   dataXY->add(*w->set("obsXY"));
 
   // now we need new model configs, with PDF="model"
   ModelConfig b_modelXY("B_modelXY", w);
   b_modelXY.SetPdf(*w->pdf("model")); // IMPORTANT
   b_modelXY.SetObservables(*w->set("obsXY"));
   b_modelXY.SetParametersOfInterest(*w->set("poi"));
   w->var("s")->setVal(0.0); // IMPORTANT
   b_modelXY.SetSnapshot(*w->set("poi"));
 
   // create the alternate (signal+background) ModelConfig with s=50
   ModelConfig sb_modelXY("S+B_modelXY", w);
   sb_modelXY.SetPdf(*w->pdf("model")); // IMPORTANT
   sb_modelXY.SetObservables(*w->set("obsXY"));
   sb_modelXY.SetParametersOfInterest(*w->set("poi"));
   w->var("s")->setVal(50.0); // IMPORTANT
   sb_modelXY.SetSnapshot(*w->set("poi"));
 
   // without this print, their can be a crash when using PROOF.  Strange.
   //  w->Print();
 
   // Test statistics like the profile likelihood ratio
   // (or the ratio of profiled likelihoods (Tevatron) or the MLE for s)
   // will now work, since the nuisance parameter b is constrained by y.
   // ratio of alt and null likelihoods with background yield profiled.
   //
   // NOTE: These are slower because they have to run fits for each toy
 
   // Tevatron-style Ratio of profiled likelihoods
   // $Q_Tev = -log L(s=0,\hat\hat{b})/L(s=50,\hat\hat{b})$
   RatioOfProfiledLikelihoodsTestStat ropl(*b_modelXY.GetPdf(), *sb_modelXY.GetPdf(), sb_modelXY.GetSnapshot());
   ropl.SetSubtractMLE(false);
 
   // profile likelihood where alternate is best fit value of signal yield
   // $\lambda(0) = -log L(s=0,\hat\hat{b})/L(\hat{s},\hat{b})$
   ProfileLikelihoodTestStat profll(*b_modelXY.GetPdf());
 
   // just use the maximum likelihood estimate of signal yield
   // $MLE = \hat{s}$
   MaxLikelihoodEstimateTestStat mlets(*sb_modelXY.GetPdf(), *w->var("s"));
 
   // However, it is less clear how to justify the prior used in randomizing
   // the nuisance parameters (since that is a property of the ensemble,
   // and y is a property of each toy pseudo experiment.  In that case,
   // one probably wants to consider a different y0 which will be held
   // constant and the prior $\pi(b) = Pois(y0 | tau b) * \eta(b)$.
   w->factory("y0[100]");
   w->factory("Gamma::gamma_y0(b,sum::temp0(y0,1),1,0)");
   w->factory("Gaussian::gauss_prior_y0(b,y0, expr::sqrty0('sqrt(y0)',y0))");
 
   // HYBRID CALCULATOR
   HybridCalculator hc3(*dataXY, sb_modelXY, b_modelXY);
   ToyMCSampler *toymcs3 = (ToyMCSampler *)hc3.GetTestStatSampler();
   toymcs3->SetNEventsPerToy(1);
   toymcs3->SetTestStatistic(&slrts);
   hc3.SetToys(ntoys, ntoys / nToysRatio);
   hc3.ForcePriorNuisanceAlt(*w->pdf("gamma_y0"));
   hc3.ForcePriorNuisanceNull(*w->pdf("gamma_y0"));
   // if you wanted to use the ad hoc Gaussian prior instead
   // ~~~{.cpp}
   // hc3.ForcePriorNuisanceAlt(*w->pdf("gauss_prior_y0"));
   // hc3.ForcePriorNuisanceNull(*w->pdf("gauss_prior_y0"));
   // ~~~
 
   // choose fit-based test statistic
   toymcs3->SetTestStatistic(&profll);
   // toymcs3->SetTestStatistic(&ropl);
   // toymcs3->SetTestStatistic(&mlets);
 
   // enable proof
   if (pc)
      toymcs3->SetProofConfig(pc);
 
   // these lines save current msg level and then kill any messages below ERROR
   RooMsgService::instance().setGlobalKillBelow(RooFit::ERROR);
   // Get the result
   HypoTestResult *r3 = hc3.GetHypoTest();
   cout << "-----------------------------------------" << endl;
   cout << "Part 6" << endl;
   r3->Print();
   t.Stop();
   t.Print();
   t.Reset();
   t.Start();
   RooMsgService::instance().setGlobalKillBelow(msglevel);
 
   c->cd(4);
   c->GetPad(4)->SetLogy();
   HypoTestPlot *p3 = new HypoTestPlot(*r3, 50); // 50 bins
   p3->Draw();
 
   c->SaveAs("zbi.pdf");
 
   // -----------------------------------------
   // OUTPUT W/O PROOF (2.66 GHz Intel Core i7)
   // =========================================
 
   // -----------------------------------------
   // Part 2
   // Hybrid p-value from direct integration = 0.00094165
   // Z_Gamma Significance  = 3.10804
   // -----------------------------------------
   //
   // Part 3
   // Z_Bi p-value (analytic): 0.00094165
   // Z_Bi significance (analytic): 3.10804
   // Real time 0:00:00, CP time 0.610
 
   // -----------------------------------------
   // Part 4
   // Results HybridCalculator_result:
   // - Null p-value = 0.00115 +/- 0.000228984
   // - Significance = 3.04848 sigma
   // - Number of S+B toys: 1000
   // - Number of B toys: 20000
   // - Test statistic evaluated on data: 150
   // - CL_b: 0.99885 +/- 0.000239654
   // - CL_s+b: 0.476 +/- 0.0157932
   // - CL_s: 0.476548 +/- 0.0158118
   // Real time 0:00:07, CP time 7.620
 
   // -----------------------------------------
   // Part 5
   // Results HybridCalculator_result:
   // - Null p-value = 0.0009 +/- 0.000206057
   // - Significance = 3.12139 sigma
   // - Number of S+B toys: 1000
   // - Number of B toys: 20000
   // - Test statistic evaluated on data: 10.8198
   // - CL_b: 0.9991 +/- 0.000212037
   // - CL_s+b: 0.465 +/- 0.0157726
   // - CL_s: 0.465419 +/- 0.0157871
   // Real time 0:00:34, CP time 34.360
 
   // -----------------------------------------
   // Part 6
   // Results HybridCalculator_result:
   // - Null p-value = 0.000666667 +/- 0.000149021
   // - Significance = 3.20871 sigma
   // - Number of S+B toys: 1000
   // - Number of B toys: 30000
   // - Test statistic evaluated on data: 5.03388
   // - CL_b: 0.999333 +/- 0.000149021
   // - CL_s+b: 0.511 +/- 0.0158076
   // - CL_s: 0.511341 +/- 0.0158183
   // Real time 0:05:06, CP time 306.330
 
   // ---------------------------------------------------------
   // OUTPUT w/ PROOF (2.66 GHz Intel Core i7, 4 virtual cores)
   // =========================================================
 
   // -----------------------------------------
   // Part 5
   // Results HybridCalculator_result:
   // - Null p-value = 0.00075 +/- 0.000173124
   // - Significance = 3.17468 sigma
   // - Number of S+B toys: 1000
   // - Number of B toys: 20000
   // - Test statistic evaluated on data: 10.8198
   // - CL_b: 0.99925 +/- 0.000193577
   // - CL_s+b: 0.454 +/- 0.0157443
   // - CL_s: 0.454341 +/- 0.0157564
   // Real time 0:00:16, CP time 0.990
 
   // -----------------------------------------
   // Part 6
   // Results HybridCalculator_result:
   // - Null p-value = 0.0007 +/- 0.000152699
   // - Significance = 3.19465 sigma
   // - Number of S+B toys: 1000
   // - Number of B toys: 30000
   // - Test statistic evaluated on data: 5.03388
   // - CL_b: 0.9993 +/- 0.000152699
   // - CL_s+b: 0.518 +/- 0.0158011
   // - CL_s: 0.518363 +/- 0.0158124
   // Real time 0:01:25, CP time 0.580
 
   // ----------------------------------
   // Comparison
   // ----------------------------------
   // LEPStatToolsForLHC
   // https://plone4.fnal.gov:4430/P0/phystat/packages/0703002
   // Uses Gaussian prior
   // CL_b = 6.218476e-04, Significance = 3.228665 sigma
   //
   // ----------------------------------
   // Comparison
   // ----------------------------------
   // Asymptotic
   // From the value of the profile likelihood ratio (5.0338)
   // The significance can be estimated using Wilks's theorem
   // significance = sqrt(2*profileLR) = 3.1729 sigma
}