HybridInstructional.C

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//+  example demostrating usage of HybridCalcultor 
/*
HybridInstructional

Authors: Kyle Cranmer, Wouter Verkerke, and Sven Kreiss
date  May 2010 Part 1-3 
date  Dec 2010 Part 4-6

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 \eta(b) then one can obtain a posterior from the auxiliary measurement
\pi(b) = \eta(b) * Pois(y|tau*b).  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
Authors: 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
*/

#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() {
  
  // 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.

#ifdef __CINT__
  cout << "DO NOT RUN WITH CINT: we are using a custom test statistic ";
  cout << "which requires that this tutorial must be compiled ";
  cout << "with ACLIC" << endl;
  return;
#endif


  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[100,0,300]))");
  w->factory("Poisson::py(y[100,0,500],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);
  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 likleihood 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 equivlaent 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(20000,1000); 
  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(20000,1000); 
  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(30000,1000); 
  hc3.ForcePriorNuisanceAlt(*w->pdf("gamma_y0"));
  hc3.ForcePriorNuisanceNull(*w->pdf("gamma_y0"));
  // if you wanted to use the ad hoc Gaussian prior instead
  //  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
  ///////////////////////////////////////////
  // Asymptotics
  // 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


}

---