TRolke.cxx

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// @(#)root/physics:$Id$
// Author: Jan Conrad
 
/** \class TRolke
    \ingroup Physics
 This class computes confidence intervals for the rate of a Poisson
 process in the presence of uncertain background and/or efficiency.
 
 The treatment and the resulting limits are fully frequentist. The
 limit calculations make use of the profile likelihood method.
 
\author Jan Conrad (CERN) 2004, Updated: Johan Lundberg (CERN) 2009
 
 For a full list of methods and their syntax, and build instructions,
 consult the header file TRolke.h.
 
 Examples/tutorials are found in the separate file Rolke.C
 
### TRolke implements the following Models
 
 The signal is always assumed to be Poisson, with the following
 combinations of models of background and detection efficiency:
 
 If unsure, first consider model 3, 4 or 5.
 
1: SetPoissonBkgBinomEff(x,y,z,tau,m)
~~~
   Background: Poisson
   Efficiency: Binomial
~~~
   when the background is simultaneously measured
   from sidebands (or MC), and
   the signal efficiency was determined from Monte Carlo
 
2: SetPoissonBkgGaussEff(x,y,em,sde,tau)
~~~
   Background: Poisson
   Efficiency: Gaussian
~~~
   when the background is simultaneously measured
   from sidebands (or MC), and
   the efficiency is modeled as Gaussian
 
3: SetGaussBkgGaussEff(x,bm,em,sde,sdb)
~~~
   Background: Gaussian
   Efficiency: Gaussian
~~~
   when background and efficiency can both be
   modeled as Gaussian.
 
4: SetPoissonBkgKnownEff(x,y,tau,e)
~~~
   Background: Poisson
   Efficiency: Known
~~~
   when the background is simultaneously measured
   from sidebands (or MC).
 
5: SetGaussBkgKnownEff(x,bm,sdb,e)
~~~
   Background: Gaussian
   Efficiency: Known
~~~
   when background is Gaussian
 
6: SetKnownBkgBinomEff(x,z,b,m)
~~~
   Background: Known
   Efficiency: Binomial
~~~
   when signal efficiency was determined from Monte Carlo
 
7: SetKnownBkgGaussEff(x,em,sde,b)
~~~
   Background: Known
   Efficiency: Gaussian
~~~
   when background is known and efficiency Gaussian
 
### Parameters and further explanation
 
#### For all models:
~~~
   x = number of observed events in the experiment
~~~
   Efficiency (e or em) is the detection probability for signal.
   A low efficiency hence generally means weaker limits.
   If the efficiency of an experiment (with analysis cuts) is
   dealt with elsewhere, em or e can be set to one.
 
#### For Poisson background measurements (sideband or MC):
~~~
   y = number of observed events in background region
   tau =
       Either: the ratio between signal and background region
       in case background is observed.
       Or: the ratio between observed and simulated live-time
       in case background is determined from MC.
~~~
#### For Gaussian efficiency or background:
~~~
   bm  = estimate of the background
   sdb = corresponding standard deviation
 
   em  = estimate of the efficiency
   sde = corresponding standard deviation
~~~
   If the efficiency scale of dealt with elsewhere,
   set em to 1 and sde to the relative uncertainty.
 
#### For Binomial signal efficiency:
~~~
    m = number of MC events generated
    z = number of MC events observed
~~~
#### For the case of known background expectation or known efficiency:
~~~
    e = true efficiency (considered known)
    b = background expectation value (considered known)
~~~
 
 The confidence level (CL) is set either at construction
 time or with either of SetCL or SetCLSigmas
 
 The TRolke method is very similar to the one used in MINUIT (MINOS).
 
 Two options are offered to deal with cases where the maximum likelihood
 estimate (MLE) is not in the physical region. Version "bounded likelihood"
 is the one used by MINOS if bounds for the physical region are chosen.
 Unbounded likelihood (the default) allows the MLE to be in the
 unphysical region. It has however better coverage.
 For more details consult the reference (see below).
 
 For a description of the method and its properties:
 
 W.Rolke, A. Lopez, J. Conrad and Fred James
 "Limits and Confidence Intervals in presence of nuisance parameters"
  http://lanl.arxiv.org/abs/physics/0403059
  Nucl.Instrum.Meth.A551:493-503,2005
 
#### Should I use TRolke, TFeldmanCousins, TLimit?
 
   1. Does TRolke make TFeldmanCousins obsolete?
      Certainly not. TFeldmanCousins is the fully frequentist construction and
      should be used in case of no (or negligible) uncertainties. It is however
      not capable of treating uncertainties in nuisance parameters. In other
      words, it does not handle background expectations or signal efficiencies
      which are known only with some limited accuracy.
      TRolke is designed for this case and it is shown in the reference above
      that it has good coverage properties for most cases, and can be used
      where FeldmannCousins can't.
 
   2. What are the advantages of TRolke over TLimit?
      TRolke is fully frequentist. TLimit treats nuisance parameters Bayesian.
      For a coverage study of a Bayesian method refer to
      physics/0408039 (Tegenfeldt & J.C). However, this note studies
      the coverage of Feldman&Cousins with Bayesian treatment of nuisance
      parameters. To make a long story short: using the Bayesian method you
      might introduce a small amount of over-coverage (though I haven't shown it
      for TLimit). On the other hand, coverage of course is a not so interesting
      when you consider yourself a Bayesian.
*/
 
#include "TRolke.h"
#include "TMath.h"
#include <iostream>
 
ClassImp(TRolke);
 
////////////////////////////////////////////////////////////////////////////////
/// Constructor with optional Confidence Level argument.
/// 'option' is not used.
 
TRolke::TRolke(Double_t CL, Option_t * /*option*/)
:  fCL(CL),
   fUpperLimit(0.0),
   fLowerLimit(0.0),
   fBounding(false),  // true gives bounded likelihood
   fNumWarningsDeprecated1(0),
   fNumWarningsDeprecated2(0)
{
   SetModelParameters();
}
 
////////////////////////////////////////////////////////////////////////////////
/// Destructor.
 
TRolke::~TRolke()
{
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 1: Background - Poisson, Efficiency - Binomial
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - z   : number of MC events observed
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - m   : number of MC events generated
 
void TRolke::SetPoissonBkgBinomEff(Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         y  ,       //   Int_t y,
         z  ,       //   Int_t z,
         0  ,       //   Double_t bm,
         0  ,       //   Double_t em,
         0  ,       //   Double_t e,
         1  ,       //   Int_t mid,
         0  ,       //   Double_t sde,
         0  ,       //   Double_t sdb,
         tau,       //   Double_t tau,
         0  ,       //   Double_t b,
         m);        //   Int_t m
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 2: Background - Poisson, Efficiency - Gaussian
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - em  : estimate of the efficiency
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - sde : efficiency estimate's standard deviation
 
void TRolke::SetPoissonBkgGaussEff(Int_t x, Int_t y, Double_t em, Double_t tau, Double_t sde)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         y  ,       //   Int_t y,
         0  ,       //   Int_t z,
         0  ,       //   Double_t bm,
         em ,       //   Double_t em,
         0  ,       //   Double_t e,
         2  ,       //   Int_t mid,
         sde,       //   Double_t sde,
         0  ,       //   Double_t sdb,
         tau,       //   Double_t tau,
         0  ,       //   Double_t b,
         0);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 3: Background - Gaussian, Efficiency - Gaussian (x,bm,em,sde,sdb)
///   - x   : number of observed events in the experiment
///   - bm  : estimate of the background
///   - em  : estimate of the efficiency
///   - sde : efficiency estimate's standard deviation
///   - sdb : background estimate's standard deviation
 
void TRolke::SetGaussBkgGaussEff(Int_t x, Double_t bm, Double_t em, Double_t sde, Double_t sdb)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         0  ,       //   Int_t y,
         0  ,       //   Int_t z,
         bm ,       //   Double_t bm,
         em ,       //   Double_t em,
         0  ,       //   Double_t e,
         3  ,       //   Int_t mid,
         sde,       //   Double_t sde,
         sdb,       //   Double_t sdb,
         0  ,       //   Double_t tau,
         0  ,       //   Double_t b,
         0);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 4: Background - Poisson, Efficiency - known     (x,y,tau,e)
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - e   : true efficiency (considered known)
 
void TRolke::SetPoissonBkgKnownEff(Int_t x, Int_t y, Double_t tau, Double_t e)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         y  ,       //   Int_t y,
         0  ,       //   Int_t z,
         0  ,       //   Double_t bm,
         0  ,       //   Double_t em,
         e  ,       //   Double_t e,
         4  ,       //   Int_t mid,
         0  ,       //   Double_t sde,
         0  ,       //   Double_t sdb,
         tau,       //   Double_t tau,
         0  ,       //   Double_t b,
         0);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 5: Background - Gaussian, Efficiency - known    (x,bm,sdb,e
///   - x   : number of observed events in the experiment
///   - bm  : estimate of the background
///   - sdb : background estimate's standard deviation
///   - e   : true efficiency (considered known)
 
void TRolke::SetGaussBkgKnownEff(Int_t x, Double_t bm, Double_t sdb, Double_t e)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         0  ,       //   Int_t y,
         0  ,       //   Int_t z,
         bm ,       //   Double_t bm,
         0  ,       //   Double_t em,
         e  ,       //   Double_t e,
         5  ,       //   Int_t mid,
         0  ,       //   Double_t sde,
         sdb,       //   Double_t sdb,
         0  ,       //   Double_t tau,
         0  ,       //   Double_t b,
         0);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 6: Background - known, Efficiency - Binomial    (x,z,m,b)
///   - x   : number of observed events in the experiment
///   - z   : number of MC events observed
///   - m   : number of MC events generated
///   - b   : background expectation value (considered known)
 
void TRolke::SetKnownBkgBinomEff(Int_t x, Int_t z, Int_t m, Double_t b)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         0  ,       //   Int_t y
         z  ,       //   Int_t z,
         0  ,       //   Double_t bm,
         0  ,       //   Double_t em,
         0  ,       //   Double_t e,
         6  ,       //   Int_t mid,
         0  ,       //   Double_t sde,
         0  ,       //   Double_t sdb,
         0  ,       //   Double_t tau,
         b  ,       //   Double_t b,
         m);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Model 7: Background - known, Efficiency - Gaussian    (x,em,sde,b)
///   - x   : number of observed events in the experiment
///   - em  : estimate of the efficiency
///   - sde : efficiency estimate's standard deviation
///   - b   : background expectation value (considered known)
 
void TRolke::SetKnownBkgGaussEff(Int_t x, Double_t em, Double_t sde, Double_t b)
{
   SetModelParameters(
         x  ,       //   Int_t x,
         0  ,       //   Int_t y
         0  ,       //   Int_t z,
         0  ,       //   Double_t bm,
         em ,       //   Double_t em,
         0  ,       //   Double_t e,
         7  ,       //   Int_t mid,
         sde,       //   Double_t sde,
         0  ,       //   Double_t sdb,
         0  ,       //   Double_t tau,
         b  ,       //   Double_t b,
         0);        //   Int_t m
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculate and get the upper and lower limits for the pre-specified model.
 
bool TRolke::GetLimits(Double_t& low, Double_t& high)
{
   if ((f_mid<1)||(f_mid>7)) {
      std::cerr << "TRolke - Error: Model id "<< f_mid<<std::endl;
      if (f_mid<1) {
         std::cerr << "TRolke - Please specify a model with e.g. 'SetGaussBkgGaussEff' (read the docs in Rolke.cxx )"<<std::endl;
      }
      return false;
   }
 
   ComputeInterval(f_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
   low = fLowerLimit;
   high = fUpperLimit;
   if (low < high) {
      return true;
   }else{
      std::cerr << "TRolke - Warning: no limits found" <<std::endl;
      return false;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculate and get upper limit for the pre-specified model.
 
Double_t TRolke::GetUpperLimit()
{
   Double_t low(0), high(0);
   GetLimits(low,high);
   return fUpperLimit;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculate and get lower limit for the pre-specified model.
 
Double_t TRolke::GetLowerLimit()
{
   Double_t low(0), high(0);
   GetLimits(low,high);
   return fLowerLimit;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Return a simple background value (estimate/truth) given the pre-specified model.
 
Double_t TRolke::GetBackground()
{
   Double_t background = 0;
   switch (f_mid) {
      case 1:
      case 2:
      case 4:
         background = f_y / f_tau;
         break;
      case 3:
      case 5:
         background = f_bm;
         break;
      case 6:
      case 7:
         background = f_b;
         break;
      default:
         std::cerr << "TRolke::GetBackground(): Model NR: " <<
         f_mid << " unknown"<<std::endl;
         return 0;
   }
   return background;
}
 
////////////////////////////////////////////////////////////////////////////////
/// get the upper and lower average limits based on the specified model.
/// No uncertainties are considered for the Poisson weights in the averaging sum
 
bool TRolke::GetSensitivity(Double_t& low, Double_t& high, Double_t pPrecision)
{
   Double_t background = GetBackground();
 
   Double_t weight = 0;
   Double_t weightSum = 0;
 
   int loop_x = 0;
 
   while (1) {
      ComputeInterval(loop_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
      weight = TMath::PoissonI(loop_x, background);
      low += fLowerLimit * weight;
      high += fUpperLimit * weight;
      weightSum += weight;
      if (loop_x > (background + 1)) { // don't stop too early
         if ((weightSum > (1 - pPrecision)) || (weight < 1e-12)) break;
      }
      loop_x++;
   }
   low /= weightSum;
   high /= weightSum;
 
   return (low < high); // could also add more detailed test
}
 
////////////////////////////////////////////////////////////////////////////////
/// get the upper and lower limits for the outcome corresponding to
/// a given quantile.
/// For integral=0.5 this gives the median limits
/// in repeated experiments. The returned out_x is the corresponding
/// (e.g. median) value of x.
/// No uncertainties are considered for the Poisson weights when calculating
/// the Poisson integral.
 
bool TRolke::GetLimitsQuantile(Double_t& low, Double_t& high, Int_t& out_x, Double_t integral)
{
   Double_t background = GetBackground();
   Double_t weight = 0;
   Double_t weightSum = 0;
   Int_t loop_x = 0;
 
   while (1) {
      weight = TMath::PoissonI(loop_x, background);
      weightSum += weight;
      if (weightSum >= integral) {
         break;
      }
      loop_x++;
   }
 
   out_x = loop_x;
 
   ComputeInterval(loop_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
   low = fLowerLimit;
   high = fUpperLimit;
   return (low < high); // could also add more detailed test
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// get the upper and lower limits for the most likely outcome.
/// The returned out_x is the corresponding value of x
/// No uncertainties are considered for the Poisson weights when finding ML.
 
bool TRolke::GetLimitsML(Double_t& low, Double_t& high, Int_t& out_x)
{
   Double_t background = GetBackground();
 
   Int_t loop_x = 0; // this can be optimized if needed.
   Int_t loop_max = 1000 + (Int_t)background; //     --||--
 
   Double_t max = TMath::PoissonI(loop_x, background);
   while (loop_x <= loop_max) {
      if (TMath::PoissonI(loop_x + 1, background) < max) {
         break;
      }
      loop_x++;
      max = TMath::PoissonI(loop_x, background);
   }
   if (loop_x >= loop_max) {
      std::cout << "internal error finding maximum of distribution" << std::endl;
      return false;
   }
 
   out_x = loop_x;
 
   ComputeInterval(loop_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
   low = fLowerLimit;
   high = fUpperLimit;
   return (low < high); // could also add more detailed test
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// get the value of x corresponding to rejection of the null hypothesis.
/// This means a lower limit >0 with the pre-specified Confidence Level.
/// Optionally give maxtry; the maximum value of x to try. Of not, or if
/// maxtry<0 an automatic mode is used.
 
bool TRolke::GetCriticalNumber(Int_t& ncrit, Int_t maxtry)
{
   Double_t background = GetBackground();
 
   int j = 0;
   int rolke_ncrit = -1;
   int maxj =maxtry ;
   if(maxtry<1){
     maxj = 1000 + (Int_t)background; // max value to try
   }
   for (j = 0;j < maxj;j++) {
      Int_t rolke_x = j;
      ComputeInterval(rolke_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
      double rolke_ll = fLowerLimit;
      if (rolke_ll > 0) {
         rolke_ncrit = j;
         break;
      }
   }
 
   if (rolke_ncrit == -1) {
     std::cerr << "TRolke GetCriticalNumber : Error: problem finding rolke inverse. Specify a larger maxtry value. maxtry was: " << maxj << ". highest x considered was j "<< j<< std::endl;
      ncrit = -1;
      return false;
   } else {
      ncrit = rolke_ncrit;
      return true;
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Deprecated name for SetBounding.
 
void TRolke::SetSwitch(bool bnd) {
   if(fNumWarningsDeprecated1<2){
      std::cerr << "*******************************************" <<std::endl;
      std::cerr << "TRolke - Warning: 'SetSwitch' is depricated and may be removed from future releases:" <<std::endl;
      std::cerr << " - Use 'SetBounding' instead "<<std::endl;
      std::cerr << "*******************************************" <<std::endl;
      fNumWarningsDeprecated1++;
   }
   SetBounding(bnd);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Dump internals. Print members.
 
void TRolke::Print(Option_t*) const {
   std::cout << "*******************************************" <<std::endl;
   std::cout << "* TRolke::Print() - dump of internals:                " <<std::endl;
   std::cout << "*"<<std::endl;
   std::cout << "* model id, mid = "<<f_mid <<std::endl;
   std::cout << "*"<<std::endl;
   std::cout << "*             x = "<<f_x   <<std::endl;
   std::cout << "*            bm = "<<f_bm  <<std::endl;
   std::cout << "*            em = "<<f_em  <<std::endl;
   std::cout << "*           sde = "<<f_sde <<std::endl;
   std::cout << "*           sdb = "<<f_sdb <<std::endl;
   std::cout << "*             y = "<<f_y   <<std::endl;
   std::cout << "*           tau = "<<f_tau <<std::endl;
   std::cout << "*             e = "<<f_e   <<std::endl;
   std::cout << "*             b = "<<f_b   <<std::endl;
   std::cout << "*             m = "<<f_m   <<std::endl;
   std::cout << "*             z = "<<f_z   <<std::endl;
   std::cout << "*"<<std::endl;
   std::cout << "*            CL = "<<fCL <<std::endl;
   std::cout << "*      Bounding = "<<fBounding <<std::endl;
   std::cout << "*"<<std::endl;
   std::cout << "* calculated on demand only:"<<std::endl;
   std::cout << "*   fUpperLimit = "<<fUpperLimit<<std::endl;
   std::cout << "*   fLowerLimit = "<<fLowerLimit<<std::endl;
   std::cout << "*******************************************" <<std::endl;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Deprecated and error prone model selection interface.
/// It's use is trongly discouraged. 'mid' is the model ID (1 to 7).
/// This method is provided for backwards compatibility/developer use only. */
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - z   : number of MC events observed
///   - bm  : estimate of the background
///   - em  : estimate of the efficiency
///   - e   : true efficiency (considered known)
///   - mid : internal model id (really, you should not use this method at all)
///   - sde : efficiency estimate's standard deviation
///   - sdb : background estimate's standard deviation
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - b   : background expectation value (considered known)
///   - m   : number of MC events generated
 
Double_t TRolke::CalculateInterval(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em, Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m){
   if (fNumWarningsDeprecated2<2 ) {
      std::cerr << "*******************************************" <<std::endl;
      std::cerr << "TRolke - Warning: 'CalculateInterval' is depricated and may be removed from future releases:" <<std::endl;
      std::cerr << " - Use e.g. 'SetGaussBkgGaussEff' and 'GetLimits' instead (read the docs in Rolke.cxx )"<<std::endl;
      std::cerr << "*******************************************" <<std::endl;
      fNumWarningsDeprecated2++;
   }
   SetModelParameters(
         x,
         y,
         z,
         bm,
         em,
         e,
         mid,
         sde,
         sdb,
         tau,
         b,
         m);
   return ComputeInterval(f_x, f_y, f_z, f_bm, f_em, f_e, f_mid, f_sde, f_sdb, f_tau, f_b, f_m);
}
 
////////////////////////////////////////////////////////////////////////////////
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - z   : number of MC events observed
///   - bm  : estimate of the background
///   - em  : estimate of the efficiency
///   - e   : true efficiency (considered known)
///   - mid : internal model id
///   - sde : efficiency estimate's standard deviation
///   - sdb : background estimate's standard deviation
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - b   : background expectation value (considered known)
///   - m   : number of MC events generated
 
void TRolke::SetModelParameters(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em, Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m)
{
   f_x   = x   ;
   f_y   = y   ;
   f_z   = z   ;
   f_bm  = bm  ;
   f_em  = em  ;
   f_e   = e   ;
   f_mid = mid ;
   f_sde = sde ;
   f_sdb = sdb ;
   f_tau = tau ;
   f_b   = b   ;
   f_m   = m   ;
}
 
void TRolke::SetModelParameters()
{
/* Clear internal model */
   SetModelParameters(0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0);
   f_mid=0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// ComputeInterval, the internals.
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - z   : number of MC events observed
///   - bm  : estimate of the background
///   - em  : estimate of the efficiency
///   - e   : true efficiency (considered known)
///   - mid : internal model id (really, you should not use this method at all)
///   - sde : efficiency estimate's standard deviation
///   - sdb : background estimate's standard deviation
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - b   : background expectation value (considered known)
///   - m   : number of MC events generated
 
Double_t TRolke::ComputeInterval(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em, Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m)
{
   //calculate interval
   Int_t done = 0;
   Double_t limit[2];
 
   limit[1] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
 
   if (limit[1] > 0) {
      done = 1;
   }
 
   if (! fBounding) {
 
      Int_t trial_x = x;
 
      while (done == 0) {
         trial_x++;
         limit[1] = Interval(trial_x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
         if (limit[1] > 0) done = 1;
      }
   }
 
   return limit[1];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Internal helper function 'Interval'
///   - x   : number of observed events in the experiment
///   - y   : number of observed events in background region
///   - z   : number of MC events observed
///   - bm  : estimate of the background
///   - em  : estimate of the efficiency
///   - e   : true efficiency (considered known)
///   - mid : internal model id (really, you should not use this method at all)
///   - sde : efficiency estimate's standard deviation
///   - sdb : background estimate's standard deviation
///   - tau : ratio parameter (read TRolke.cxx for details)
///   - b   : background expectation value (considered known)
///   - m   : number of MC events generated
 
Double_t TRolke::Interval(Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em, Double_t e, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m)
{
   Double_t dchi2 = TMath::ChisquareQuantile(fCL, 1);
   Double_t tempxy[2], limits[2] = {0, 0};
   Double_t slope, fmid, low, flow, high, fhigh, test, ftest, mu0, maximum, target, l, f0;
   Double_t med = 0;
   Double_t maxiter = 1000, acc = 0.00001;
   Int_t i;
   Int_t bp = 0;
 
   if ((mid != 3) && (mid != 5)) bm = y;
   if ((mid == 3) || (mid == 5)) {
      if (bm == 0) bm = 0.00001;
   }
 
   if ((mid == 6) || (mid == 7)) {
      if (bm == 0) bm = 0.00001;
   }
 
   if ((mid <= 2) || (mid == 4)) bp = 1;
 
 
   if (bp == 1 && x == 0 && bm > 0) {
      for (i = 0; i < 2; i++) {
         x++;
         tempxy[i] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
      }
 
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      limits[0] = 0.0;
      if (limits[1] < 0) limits[1] = 0.0;
      goto done;
   }
 
   if (bp != 1 && x == 0) {
 
      for (i = 0; i < 2; i++) {
         x++;
         tempxy[i] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
      }
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      limits[0] = 0.0;
      if (limits[1] < 0) limits[1] = 0.0;
      goto done;
   }
 
   if (bp != 1  && bm == 0) {
      for (i = 0; i < 2; i++) {
         bm++;
         limits[1] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
         tempxy[i] = limits[1];
      }
      slope = tempxy[1] - tempxy[0];
      limits[1] = tempxy[0] - slope;
      if (limits[1] < 0) limits[1] = 0;
      goto done;
   }
 
   if (x == 0 && bm == 0) {
      x++;
      bm++;
      limits[1] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
      tempxy[0] = limits[1];
      x  = 1;
      bm = 2;
      limits[1] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
      tempxy[1] = limits[1];
      x  = 2;
      bm = 1;
      limits[1] = Interval(x, y, z, bm, em, e, mid, sde, sdb, tau, b, m);
      limits[1] = 3 * tempxy[0] - tempxy[1] - limits[1];
      if (limits[1] < 0) limits[1] = 0;
      goto done;
   }
 
   mu0 = Likelihood(0, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 1);
   maximum = Likelihood(0, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 2);
   test = 0;
   f0 = Likelihood(test, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 3);
   if (fBounding) {
      if (mu0 < 0) maximum = f0;
   }
 
   target = maximum - dchi2;
   if (f0 > target) {
      limits[0] = 0;
   } else {
      if (mu0 < 0) {
         limits[0] = 0;
         limits[1] = 0;
      }
 
      low   = 0;
      flow  = f0;
      high  = mu0;
      fhigh = maximum;
      for (i = 0; i < maxiter; i++) {
         l = (target - fhigh) / (flow - fhigh);
         if (l < 0.2) l = 0.2;
         if (l > 0.8) l = 0.8;
         med = l * low + (1 - l) * high;
         if (med < 0.01) {
            limits[1] = 0.0;
            goto done;
         }
         fmid = Likelihood(med, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 3);
         if (fmid > target) {
            high  = med;
            fhigh = fmid;
         } else {
            low  = med;
            flow = fmid;
         }
         if ((high - low) < acc*high) break;
      }
      limits[0] = med;
   }
 
   if (mu0 > 0) {
      low  = mu0;
      flow = maximum;
   } else {
      low  = 0;
      flow = f0;
   }
 
   test = low + 1 ;
   ftest = Likelihood(test, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 3);
   if (ftest < target) {
      high  = test;
      fhigh = ftest;
   } else {
      slope = (ftest - flow) / (test - low);
      high  = test + (target - ftest) / slope;
      fhigh = Likelihood(high, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 3);
   }
 
   for (i = 0; i < maxiter; i++) {
      l = (target - fhigh) / (flow - fhigh);
      if (l < 0.2) l = 0.2;
      if (l > 0.8) l = 0.8;
      med  = l * low + (1. - l) * high;
      fmid = Likelihood(med, x, y, z, bm, em, mid, sde, sdb, tau, b, m, 3);
 
      if (fmid < target) {
         high  = med;
         fhigh = fmid;
      } else {
         low  = med;
         flow = fmid;
      }
 
      if (high - low < acc*high) break;
   }
 
   limits[1] = med;
 
done:
 
   // apply known efficiency
   if ((mid == 4) || (mid == 5)) {
      limits[0] /= e;
      limits[1] /= e;
   }
 
   fUpperLimit = limits[1];
   fLowerLimit = TMath::Max(limits[0], 0.0);
 
   return limits[1];
}
 
////////////////////////////////////////////////////////////////////////////////
/// Internal helper function
/// Chooses between the different profile likelihood functions to use for the
/// different models.
/// Returns evaluation of the profile likelihood functions.
 
Double_t TRolke::Likelihood(Double_t mu, Int_t x, Int_t y, Int_t z, Double_t bm, Double_t em, Int_t mid, Double_t sde, Double_t sdb, Double_t tau, Double_t b, Int_t m, Int_t what)
{
   switch (mid) {
      case 1:
         return EvalLikeMod1(mu, x, y, z, tau, m, what);
      case 2:
         return EvalLikeMod2(mu, x, y, em, sde, tau, what);
      case 3:
         return EvalLikeMod3(mu, x, bm, em, sde, sdb, what);
      case 4:
         return EvalLikeMod4(mu, x, y, tau, what);
      case 5:
         return EvalLikeMod5(mu, x, bm, sdb, what);
      case 6:
         return EvalLikeMod6(mu, x, z, b, m, what);
      case 7:
         return EvalLikeMod7(mu, x, em, sde, b, what);
      default:
         std::cerr << "TRolke::Likelihood(...): Model NR: " <<
         f_mid << " unknown"<<std::endl;
         return 0;
   }
 
   return 0;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 1:
/// Poisson background/ Binomial Efficiency
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod1(Double_t mu, Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m, Int_t what)
{
   Double_t f  = 0;
   Double_t zm = Double_t(z) / m;
 
   if (what == 1) {
      f = (x - y / tau) / zm;
   }
 
   if (what == 2) {
      mu = (x - y / tau) / zm;
      Double_t b  = y / tau;
      Double_t e = zm;
      f = LikeMod1(mu, b, e, x, y, z, tau, m);
   }
 
   if (what == 3) {
      if (mu == 0) {
         Double_t b = (x + y) / (1.0 + tau);
         Double_t e = zm;
         f = LikeMod1(mu, b, e, x, y, z, tau, m);
      } else {
         Double_t e = 0;
         Double_t b = 0;
         ProfLikeMod1(mu, b, e, x, y, z, tau, m);
         f = LikeMod1(mu, b, e, x, y, z, tau, m);
      }
   }
 
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 1:
/// Poisson background/ Binomial Efficiency
 
Double_t TRolke::LikeMod1(Double_t mu, Double_t b, Double_t e, Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m)
{
   double s = e*mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
   double bg = tau*b;
   double llb =  -bg;
   if ( y > 0) llb =  y*TMath::Log( bg) - bg - LogFactorial(y);
 
   double lle = 0;  // binomial log-like
   if (z == 0)         lle = m * TMath::Log(1-e);
   else if ( z == m)   lle = m * TMath::Log(e);
   else                lle =   z * TMath::Log(e) + (m - z)*TMath::Log(1 - e) + LogFactorial(m) - LogFactorial(m-z) - LogFactorial(z);
 
   double f = 2*( lls + llb + lle);
   return f;
}
 
 
// this code is non-sense - // need to solve using Minuit
struct LikeFunction1 {
};
 
////////////////////////////////////////////////////////////////////////////////
/// Helper for calculation of estimates of efficiency and background for model 1
 
void TRolke::ProfLikeMod1(Double_t mu, Double_t &b, Double_t &e, Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m)
{
   Double_t med = 0.0, fmid;
   Int_t maxiter = 1000;
   Double_t acc = 0.00001;
   Double_t emin = ((m + mu * tau) - TMath::Sqrt((m + mu * tau) * (m + mu * tau) - 4 * mu * tau * z)) / 2 / mu / tau;
 
   Double_t low  = TMath::Max(1e-10, emin + 1e-10);
   Double_t high = 1 - 1e-10;
 
   for (Int_t i = 0; i < maxiter; i++) {
      med = (low + high) / 2.;
 
      fmid = LikeGradMod1(med, mu, x, y, z, tau, m);
 
      if (high < 0.5) acc = 0.00001 * high;
      else           acc = 0.00001 * (1 - high);
 
      if ((high - low) < acc*high) break;
 
      if (fmid > 0) low  = med;
      else         high = med;
   }
 
   e = med;
   Double_t eta = Double_t(z) / e - Double_t(m - z) / (1 - e);
 
   b = Double_t(y) / (tau - eta / mu);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Gradient model likelihood
 
Double_t TRolke::LikeGradMod1(Double_t e, Double_t mu, Int_t x, Int_t y, Int_t z, Double_t tau, Int_t m)
{
   Double_t eta, etaprime, bprime, f;
   eta = static_cast<double>(z) / e - static_cast<double>(m - z) / (1.0 - e);
   etaprime = (-1) * (static_cast<double>(m - z) / ((1.0 - e) * (1.0 - e)) + static_cast<double>(z) / (e * e));
   Double_t b = y / (tau - eta / mu);
   bprime = (b * b * etaprime) / mu / y;
   f = (mu + bprime) * (x / (e * mu + b) - 1) + (y / b - tau) * bprime + eta;
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 2:
/// Poisson background/ Gauss Efficiency
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod2(Double_t mu, Int_t x, Int_t y, Double_t em, Double_t sde, Double_t tau, Int_t what)
{
   Double_t v =  sde * sde;
   Double_t coef[4], roots[3];
   Double_t f = 0;
 
   if (what == 1) {
      f = (x - y / tau) / em;
   }
 
   if (what == 2) {
      mu = (x - y / tau) / em;
      Double_t b = y / tau;
      Double_t e = em;
      f = LikeMod2(mu, b, e, x, y, em, tau, v);
   }
 
   if (what == 3) {
      if (mu == 0) {
         Double_t b = (x + y) / (1 + tau);
         Double_t e = em ;
         f = LikeMod2(mu, b, e, x, y, em, tau, v);
      } else {
         coef[3] = mu;
         coef[2] = mu * mu * v - 2 * em * mu - mu * mu * v * tau;
         coef[1] = (- x) * mu * v - mu * mu * mu * v * v * tau - mu * mu * v * em + em * mu * mu * v * tau + em * em * mu - y * mu * v;
         coef[0] = x * mu * mu * v * v * tau + x * em * mu * v - y * mu * mu * v * v + y * em * mu * v;
 
         TMath::RootsCubic(coef, roots[0], roots[1], roots[2]);
 
         Double_t e = roots[1];
         Double_t b;
         if ( v > 0) b = y / (tau + (em - e) / mu / v);
         else b = y/tau;
         f = LikeMod2(mu, b, e, x, y, em, tau, v);
      }
   }
 
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 2:
/// Poisson background/Gauss Efficiency
 
Double_t TRolke::LikeMod2(Double_t mu, Double_t b, Double_t e, Int_t x, Int_t y, Double_t em, Double_t tau, Double_t v)
{
   double s = e*mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
   double bg = tau*b;
   double llb =  -bg;
   if ( y > 0) llb =  y*TMath::Log( bg) - bg - LogFactorial(y);
   double lle = 0;
   if ( v > 0) lle = - 0.9189385 - TMath::Log(v) / 2 - (em - e)*(em - e) / v / 2;
 
   return 2*( lls + llb + lle);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 3:
/// Gauss  background/ Gauss Efficiency
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod3(Double_t mu, Int_t x, Double_t bm, Double_t em, Double_t sde, Double_t sdb, Int_t what)
{
   Double_t f = 0.;
   Double_t  v = sde * sde;
   Double_t  u = sdb * sdb;
 
   if (what == 1) {
      f = (x - bm) / em;
   }
 
 
   if (what == 2) {
      mu = (x - bm) / em;
      Double_t b  = bm;
      Double_t e  = em;
      f  = LikeMod3(mu, b, e, x, bm, em, u, v);
   }
 
 
   if (what == 3) {
      if (mu == 0.0) {
         Double_t b = ((bm - u) + TMath::Sqrt((bm - u) * (bm - u) + 4 * x * u)) / 2.;
         Double_t e = em;
         f = LikeMod3(mu, b, e, x, bm, em, u, v);
      } else {
         Double_t e = em;
         Double_t b = bm;
         if ( v > 0) {
            Double_t temp[3];
            temp[0] = mu * mu * v + u;
            temp[1] = mu * mu * mu * v * v + mu * v * u - mu * mu * v * em + mu * v * bm - 2 * u * em;
            temp[2] = mu * mu * v * v * bm - mu * v * u * em - mu * v * bm * em + u * em * em - mu * mu * v * v * x;
            e = (-temp[1] + TMath::Sqrt(temp[1] * temp[1] - 4 * temp[0] * temp[2])) / 2 / temp[0];
            b = bm - (u * (em - e)) / v / mu;
         }
         f = LikeMod3(mu, b, e, x, bm, em, u, v);
      }
   }
 
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 3:
/// Gauss background/Gauss Efficiency
 
Double_t TRolke::LikeMod3(Double_t mu, Double_t b, Double_t e, Int_t x, Double_t bm, Double_t em, Double_t u, Double_t v)
{
   double s = e*mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
   double llb =  0;
   if ( u > 0) llb = - 0.9189385 - TMath::Log(u) / 2 - (bm - b)*(bm - b) / u / 2;
   double lle = 0;
   if ( v > 0) lle = - 0.9189385 - TMath::Log(v) / 2 - (em - e)*(em - e) / v / 2;
 
   return 2*( lls + llb + lle);
 
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 4:
/// Poiss  background/Efficiency known
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod4(Double_t mu, Int_t x, Int_t y, Double_t tau, Int_t what)
{
   Double_t f = 0.0;
 
   if (what == 1) f = x - y / tau;
   if (what == 2) {
      mu = x - y / tau;
      Double_t b  = y / tau;
      f  = LikeMod4(mu, b, x, y, tau);
   }
   if (what == 3) {
      if (mu == 0.0) {
         Double_t b = Double_t(x + y) / (1 + tau);
         f = LikeMod4(mu, b, x, y, tau);
      } else {
         Double_t b = (x + y - (1 + tau) * mu + sqrt((x + y - (1 + tau) * mu) * (x + y - (1 + tau) * mu) + 4 * (1 + tau) * y * mu)) / 2 / (1 + tau);
         f = LikeMod4(mu, b, x, y, tau);
      }
   }
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 4:
/// Poiss background/Efficiency known
 
Double_t TRolke::LikeMod4(Double_t mu, Double_t b, Int_t x, Int_t y, Double_t tau)
{
   double s = mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
   double bg = tau*b;
   double llb =  -bg;
   if ( y > 0) llb =  y*TMath::Log( bg) - bg - LogFactorial(y);
 
   return 2*( lls + llb);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 5:
/// Gauss  background/Efficiency known
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod5(Double_t mu, Int_t x, Double_t bm, Double_t sdb, Int_t what)
{
   Double_t u = sdb * sdb;
   Double_t f = 0;
 
   if (what == 1) {
      f = x - bm;
   }
   if (what == 2) {
      mu = x - bm;
      Double_t b  = bm;
      f  = LikeMod5(mu, b, x, bm, u);
   }
 
   if (what == 3) {
      Double_t b = ((bm - u - mu) + TMath::Sqrt((bm - u - mu) * (bm - u - mu) - 4 * (mu * u - mu * bm - u * x))) / 2;
      f = LikeMod5(mu, b, x, bm, u);
   }
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 5:
/// Gauss background/Efficiency known
 
Double_t TRolke::LikeMod5(Double_t mu, Double_t b, Int_t x, Double_t bm, Double_t u)
{
   double s = mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
   double llb =  0;
   if ( u > 0) llb = - 0.9189385 - TMath::Log(u) / 2 - (bm - b)*(bm - b) / u / 2;
 
   return 2*( lls + llb);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 6:
/// Background known/Efficiency binomial
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod6(Double_t mu, Int_t x, Int_t z, Double_t b, Int_t m, Int_t what)
{
   Double_t coef[4], roots[3];
   Double_t f = 0.;
   Double_t zm = Double_t(z) / m;
 
   if (what == 1) {
      f = (x - b) / zm;
   }
 
   if (what == 2) {
      mu = (x - b) / zm;
      Double_t e  = zm;
      f  = LikeMod6(mu, b, e, x, z, m);
   }
   if (what == 3) {
      Double_t e;
      if (mu == 0) {
         e = zm;
      } else {
         coef[3] = mu * mu;
         coef[2] = mu * b - mu * x - mu * mu - mu * m;
         coef[1] = mu * x - mu * b + mu * z - m * b;
         coef[0] = b * z;
         TMath::RootsCubic(coef, roots[0], roots[1], roots[2]);
         e = roots[1];
      }
      f = LikeMod6(mu, b, e, x, z, m);
   }
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 6:
/// background known/ Efficiency binomial
 
Double_t TRolke::LikeMod6(Double_t mu, Double_t b, Double_t e, Int_t x, Int_t z, Int_t m)
{
   double s = e*mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
 
   double lle = 0;
   if (z == 0)        lle = m * TMath::Log(1-e);
   else if ( z == m)  lle = m * TMath::Log(e);
   else               lle =   z * TMath::Log(e) + (m - z)*TMath::Log(1 - e) + LogFactorial(m) - LogFactorial(m-z) - LogFactorial(z);
 
   return 2* (lls + lle);
}
 
 
////////////////////////////////////////////////////////////////////////////////
/// Calculates the Profile Likelihood for MODEL 7:
/// background known/Efficiency Gauss
///  - what = 1: Maximum likelihood estimate is returned
///  - what = 2: Profile Likelihood of Maximum Likelihood estimate is returned.
///  - what = 3: Profile Likelihood of Test hypothesis is returned
/// otherwise parameters as described in the beginning of the class)
 
Double_t TRolke::EvalLikeMod7(Double_t mu, Int_t x, Double_t em, Double_t sde, Double_t b, Int_t what)
{
   Double_t v = sde * sde;
   Double_t f = 0.;
 
   if (what ==  1) {
      f = (x - b) / em;
   }
 
   if (what == 2) {
      mu = (x - b) / em;
      Double_t e  = em;
      f  = LikeMod7(mu, b, e, x, em, v);
   }
 
   if (what == 3) {
      Double_t e;
      if (mu == 0) {
         e = em;
      } else {
         e = (-(mu * em - b - mu * mu * v) - TMath::Sqrt((mu * em - b - mu * mu * v) * (mu * em - b - mu * mu * v) + 4 * mu * (x * mu * v - mu * b * v + b * em))) / (- mu) / 2;
      }
      f = LikeMod7(mu, b, e, x, em, v);
   }
 
   return f;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Profile Likelihood function for MODEL 6:
/// background known/ Efficiency gaussian
 
Double_t TRolke::LikeMod7(Double_t mu, Double_t b, Double_t e, Int_t x, Double_t em, Double_t v)
{
   double s = e*mu+b;
   double lls = - s;
   if (x > 0) lls = x*TMath::Log(s) - s - LogFactorial(x);
 
   double lle = 0;
   if ( v > 0) lle = - 0.9189385 - TMath::Log(v) / 2 - (em - e)*(em - e) / v / 2;
 
   return 2*( lls + lle);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Evaluate polynomial
 
Double_t TRolke::EvalPolynomial(Double_t x, const Int_t  coef[], Int_t N)
{
   const Int_t   *p;
   p = coef;
   Double_t ans = *p++;
   Int_t i = N;
 
   do
      ans = ans * x  +  *p++;
   while (--i);
 
   return ans;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Evaluate mononomial
 
Double_t TRolke::EvalMonomial(Double_t x, const Int_t coef[], Int_t N)
{
   Double_t ans;
   const Int_t   *p;
 
   p   = coef;
   ans = x + *p++;
   Int_t i = N - 1;
 
   do
      ans = ans * x  + *p++;
   while (--i);
 
   return ans;
}
 
////////////////////////////////////////////////////////////////////////////////
/// LogFactorial function (use the logGamma function via the relation Gamma(n+1) = n!
 
Double_t TRolke::LogFactorial(Int_t n)
{
   return TMath::LnGamma(n+1);
}