BCIntegrate.cxx

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/*
 * Copyright (C) 2008, Daniel Kollar and Kevin Kroeninger.
 * All rights reserved.
 *
 * For the licensing terms see doc/COPYING.
 */

// ---------------------------------------------------------

#include "config.h"

#include "BAT/BCIntegrate.h"
#include "BAT/BCLog.h"
#include "BAT/BCMath.h"

#include <TH1D.h>
#include <TH2D.h>
#include <TMinuit.h>
#include <TRandom3.h>
#include <TTree.h>

#ifdef HAVE_CUBA_H
#include "cuba.h"
#endif

// ---------------------------------------------------------

class BCIntegrate * global_this;

// *********************************************
BCIntegrate::BCIntegrate() : BCEngineMCMC()
{
   fNvar=0;
   fNiterPerDimension = 100;
   fNSamplesPer2DBin = 100;
   fRandom = new TRandom3(0);

   fNIterationsMax    = 1000000;
   fRelativePrecision = 1e-3;

#ifdef HAVE_CUBA_H
   fIntegrationMethod = BCIntegrate::kIntCuba;
#else
   fIntegrationMethod = BCIntegrate::kIntMonteCarlo;
#endif
   fMarginalizationMethod = BCIntegrate::kMargMetropolis;
   fOptimizationMethod = BCIntegrate::kOptMinuit;
   fOptimizationMethodMode = BCIntegrate::kOptMinuit; 

   fNbins=100;

   fDataPointLowerBoundaries = 0;
   fDataPointUpperBoundaries = 0;

   fFillErrorBand = false;

   fFitFunctionIndexX = -1;
   fFitFunctionIndexY = -1;

   fErrorBandNbinsX = 100;
   fErrorBandNbinsY = 500;

   fErrorBandContinuous = true;

   fMinuit = 0;
   fMinuitArglist[0] = 20000;
   fMinuitArglist[1] = 0.01;
   fFlagIgnorePrevOptimization = false; 

   fFlagWriteMarkovChain = false;
   fMarkovChainTree = 0;

   fMarkovChainStepSize = 0.1;

   fMarkovChainAutoN = true;

   fSAT0 = 100.0;
   fSATmin = 0.1;
   fSASchedule = BCIntegrate::kSACauchy;
   fFlagWriteSAToFile = false; 
}

// *********************************************
BCIntegrate::BCIntegrate(BCParameterSet * par) : BCEngineMCMC(1)
{
   fNvar=0;
   fNiterPerDimension = 100;
   fNSamplesPer2DBin = 100;
   fRandom = new TRandom3(0);

   fNbins=100;

   this->SetParameters(par);

   fDataPointLowerBoundaries = 0;
   fDataPointUpperBoundaries = 0;

   fFillErrorBand = false;

   fFitFunctionIndexX = -1;
   fFitFunctionIndexY = -1;

   fErrorBandNbinsX = 100;
   fErrorBandNbinsY = 200;

   fErrorBandContinuous = true;

   fMinuit = 0;
   fMinuitArglist[0] = 20000;
   fMinuitArglist[1] = 0.01;
   fFlagIgnorePrevOptimization = false; 

   fFlagWriteMarkovChain = false;
   fMarkovChainTree = 0;

   fMarkovChainStepSize = 0.1;

   fMarkovChainAutoN = true;
   
   fSAT0 = 100.0;
   fSATmin = 0.1;
   fTreeSA = false; 
}

// *********************************************
BCIntegrate::~BCIntegrate()
{
   DeleteVarList();

   fx=0;

   delete fRandom;
   fRandom=0;

   if (fMinuit)
      delete fMinuit;

   int n1 = fHProb1D.size();
   if(n1>0)
   {
      for (int i=0;i<n1;i++)
         delete fHProb1D.at(i);
   }

   int n2 = fHProb2D.size();
   if(n2>0)
   {
      for (int i=0;i<n2;i++)
         delete fHProb2D.at(i);
   }
}

// *********************************************
void BCIntegrate::SetParameters(BCParameterSet * par)
{
   DeleteVarList();

   fx = par;
   fNvar = fx->size();
   fMin = new double[fNvar];
   fMax = new double[fNvar];
   fVarlist = new int[fNvar];

   this->ResetVarlist(1);

   for(int i=0;i<fNvar;i++)
   {
      fMin[i]=(fx->at(i))->GetLowerLimit();
      fMax[i]=(fx->at(i))->GetUpperLimit();
   }

   fXmetro0.clear();
   for(int i=0;i<fNvar;i++)
      fXmetro0.push_back((fMin[i]+fMax[i])/2.0);

   fXmetro1.clear();
   fXmetro1.assign(fNvar,0.);

   fMCMCBoundaryMin.clear();
   fMCMCBoundaryMax.clear();

   for(int i=0;i<fNvar;i++)
   {
      fMCMCBoundaryMin.push_back(fMin[i]);
      fMCMCBoundaryMax.push_back(fMax[i]);
      fMCMCFlagsFillHistograms.push_back(true); 
   }

   for (int i = int(fMCMCH1NBins.size()); i<fNvar; ++i)
      fMCMCH1NBins.push_back(100);

   fMCMCNParameters = fNvar;
}

// *********************************************
void BCIntegrate::SetMarkovChainInitialPosition(std::vector<double> position)
{
   int n = position.size();

   fXmetro0.clear();

   for (int i = 0; i < n; ++i)
      fXmetro0.push_back(position.at(i));
}

// *********************************************
void BCIntegrate::DeleteVarList()
{
   if(fNvar)
   {
      delete[] fVarlist;
      fVarlist=0;

      delete[] fMin;
      fMin=0;

      delete[] fMax;
      fMax=0;

      fx=0;
      fNvar=0;
   }
}

// *********************************************
void BCIntegrate::SetNbins(int nbins, int index)
{
   if (fNvar == 0)
      return;

   // check if index is in range
   if (index >= fNvar)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,"BCIntegrate::SetNbins : Index out of range.");
      return;
   }
   // set for all parameters at once
   else if (index < 0)
   {
      for (int i = 0; i < fNvar; ++i)
         this -> SetNbins(nbins, i);
      return;
   }

   // sanity check for nbins
   if (nbins <= 0)
      nbins = 10;

   fMCMCH1NBins[index] = nbins;

   return;

//    if(n<2)
//    {
//       BCLog::Out(BCLog::warning, BCLog::warning,"BCIntegrate::SetNbins. Number of bins less than 2 makes no sense. Setting to 2.");
//       n=2;
//    }
//    this -> MCMCSetH1NBins(n, -1);

   // fNbins=n;

   // fMCMCH1NBins = n;
   // fMCMCH2NBinsX = n;
   // fMCMCH2NBinsY = n;
}

// *********************************************
// void BCIntegrate::SetNbinsX(int n)
// {
//    if(n<2)
//    {
//       BCLog::Out(BCLog::warning, BCLog::warning,"BCIntegrate::SetNbins. Number of bins less than 2 makes no sense. Setting to 2.");
//       n=2;
//    }
//    fMCMCH2NBinsX = n;
// }

// *********************************************
// void BCIntegrate::SetNbinsY(int n)
// {
//    if(n<2)
//    {
//       BCLog::Out(BCLog::warning, BCLog::warning,"BCIntegrate::SetNbins. Number of bins less than 2 makes no sense. Setting to 2.");
//       n=2;
//    }
//    fNbins=n;

//    fMCMCH2NBinsY = n;
// }

// *********************************************
void BCIntegrate::SetVarList(int * varlist)
{
   for(int i=0;i<fNvar;i++)
      fVarlist[i]=varlist[i];
}

// *********************************************
void BCIntegrate::ResetVarlist(int v)
{
   for(int i=0;i<fNvar;i++)
      fVarlist[i]=v;
}

// *********************************************
double BCIntegrate::Eval(std::vector <double> x)
{
   BCLog::Out(BCLog::warning, BCLog::warning, "BCIntegrate::Eval. No function. Function needs to be overloaded.");
   return 0;
}

// *********************************************
double BCIntegrate::LogEval(std::vector <double> x)
{
   // this method should better also be overloaded
   return log(this->Eval(x));
}

// *********************************************
double BCIntegrate::EvalSampling(std::vector <double> x)
{
   BCLog::Out(BCLog::warning, BCLog::warning, "BCIntegrate::EvalSampling. No function. Function needs to be overloaded.");
   return 0;
}

// *********************************************
double BCIntegrate::LogEvalSampling(std::vector <double> x)
{
   return log(this->EvalSampling(x));
}

// *********************************************
double BCIntegrate::EvalPrint(std::vector <double> x)
{
   double val=this->Eval(x);
   BCLog::Out(BCLog::detail, BCLog::detail, Form("BCIntegrate::EvalPrint. Value: %d.", val));

   return val;
}

// *********************************************
void BCIntegrate::SetIntegrationMethod(BCIntegrate::BCIntegrationMethod method)
{
#ifdef HAVE_CUBA_H
   fIntegrationMethod = method;
#else
   BCLog::Out(BCLog::warning,BCLog::warning,"!!! This version of BAT is compiled without Cuba.");
   BCLog::Out(BCLog::warning,BCLog::warning,"    Monte Carlo Sampled Mean integration method will be used.");
   BCLog::Out(BCLog::warning,BCLog::warning,"    To be able to use Cuba integration, install Cuba and recompile BAT.");
#endif
}

// *********************************************
double BCIntegrate::Integrate()
{
   std::vector <double> parameter;
   parameter.assign(fNvar, 0.0);

   switch(fIntegrationMethod)
   {
      case BCIntegrate::kIntMonteCarlo:
         return IntegralMC(parameter);

      case BCIntegrate::kIntMetropolis:
         return this -> IntegralMetro(parameter);

      case BCIntegrate::kIntImportance:
         return this -> IntegralImportance(parameter);

      case BCIntegrate::kIntCuba:
#ifdef HAVE_CUBA_H
         return this -> CubaIntegrate();
#else
         BCLog::Out(BCLog::error,BCLog::error,"!!! This version of BAT is compiled without Cuba.");
         BCLog::Out(BCLog::error,BCLog::error,"    Use other integration methods or install Cuba and recompile BAT.");
#endif
   }

   BCLog::Out(BCLog::error, BCLog::error,
         Form("BCIntegrate::Integrate : Invalid integration method: %d", fIntegrationMethod));

   return 0;
}

// *********************************************
double BCIntegrate::IntegralMC(std::vector <double> x, int * varlist)
{
   this->SetVarList(varlist);
   return IntegralMC(x);
}

// *********************************************
double BCIntegrate::IntegralMC(std::vector <double> x)
{
   // count the variables to integrate over
   int NvarNow=0;

   for(int i = 0; i < fNvar; i++)
      if(fVarlist[i])
         NvarNow++;

   // print to log
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Running MC integation over %i dimensions.", NvarNow));
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Maximum number of iterations: %i", this->GetNIterationsMax()));
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Aimed relative precision:     %e", this->GetRelativePrecision()));

   // calculate D (the integrated volume)
   double D = 1.0;
   for(int i = 0; i < fNvar; i++)
      if (fVarlist[i])
         D = D * (fMax[i] - fMin[i]);

   // reset variables
   double pmax = 0.0;
   double sumW  = 0.0;
   double sumW2 = 0.0;
   double integral = 0.0;
   double variance = 0.0;
   double relprecision = 1.0;

   std::vector <double> randx;
   randx.assign(fNvar, 0.0);

   // reset number of iterations
   fNIterations = 0;

   // iterate while precision is not reached and the number of iterations is lower than maximum number of iterations
   while ((fRelativePrecision < relprecision && fNIterationsMax > fNIterations) || fNIterations < 10)
   {
      // increase number of iterations
      fNIterations++;

      // get random numbers
      this -> GetRandomVector(randx);

      // scale random numbers
      for(int i = 0; i < fNvar; i++)
      {
         if(fVarlist[i])
            randx[i]=fMin[i]+randx[i]*(fMax[i]-fMin[i]);
         else
            randx[i]=x[i];
      }

      // evaluate function at sampled point
      double value = this->Eval(randx);

      // add value to sum and sum of squares
      sumW  += value;
      sumW2 += value * value;

      // search for maximum probability
      if (value > pmax)
      {
         // set new maximum value
         pmax = value;

         // delete old best fit parameter values
         fBestFitParameters.clear();

         // write best fit parameters
         for(int i = 0; i < fNvar; i++)
            fBestFitParameters.push_back(randx.at(i));
      }

      // calculate integral and variance
      integral = D * sumW / fNIterations;

      if (fNIterations%10000 == 0)
      {
         variance = (1.0 / double(fNIterations)) * (D * D * sumW2 / double(fNIterations) - integral * integral);
         double error = sqrt(variance);
         relprecision = error / integral;

         BCLog::Out(BCLog::debug, BCLog::debug,
            Form("BCIntegrate::IntegralMC. Iteration %i, integral: %e +- %e.", fNIterations, integral, error));
      }
   }

   fError = variance / fNIterations;

   // print to log
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Result of integration:        %e +- %e   in %i iterations.", integral, sqrt(variance), fNIterations));
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Obtained relative precision:  %e. ", sqrt(variance) / integral));

   return integral;
}


// *********************************************
double BCIntegrate::IntegralMetro(std::vector <double> x)
{
   // print debug information
   BCLog::Out(BCLog::debug, BCLog::debug, Form("BCIntegrate::IntegralMetro. Integate over %i dimensions.", fNvar));

   // get total number of iterations
   double Niter = pow(fNiterPerDimension, fNvar);

   // print if more than 100,000 iterations
   if(Niter>1e5)
      BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Total number of iterations in Metropolis: %d.", Niter));

   // reset sum
   double sumI = 0;

   // prepare Metropolis algorithm
   std::vector <double> randx;
   randx.assign(fNvar,0.);
   InitMetro();

   // prepare maximum value
   double pmax = 0.0;

   // loop over iterations
   for(int i = 0; i <= Niter; i++)
   {
      // get random point from sampling distribution
      this -> GetRandomPointSamplingMetro(randx);

      // calculate probability at random point
      double val_f = this -> Eval(randx);

      // calculate sampling distributions at that point
      double val_g = this -> EvalSampling(randx);

      // add ratio to sum
      if (val_g > 0)
         sumI += val_f / val_g;

      // search for maximum probability
      if (val_f > pmax)
      {
         // set new maximum value
         pmax = val_f;

         // delete old best fit parameter values
         fBestFitParameters.clear();

         // write best fit parameters
         for(int i = 0; i < fNvar; i++)
            fBestFitParameters.push_back(randx.at(i));
      }

      // write intermediate results
      if((i+1)%100000 == 0)
         BCLog::Out(BCLog::debug, BCLog::debug,
            Form("BCIntegrate::IntegralMetro. Iteration %d, integral: %d.", i+1, sumI/(i+1)));
   }

   // calculate integral
   double result=sumI/Niter;

   // print debug information
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Integral is %f in %i iterations.", result, Niter));

   return result;
}

// *********************************************
double BCIntegrate::IntegralImportance(std::vector <double> x)
{
   // print debug information
   BCLog::Out(BCLog::debug, BCLog::debug, Form("BCIntegrate::IntegralImportance. Integate over %i dimensions.", fNvar));

   // get total number of iterations
   double Niter = pow(fNiterPerDimension, fNvar);

   // print if more than 100,000 iterations
   if(Niter>1e5)
      BCLog::Out(BCLog::detail, BCLog::detail, Form("BCIntegrate::IntegralImportance. Total number of iterations: %d.", Niter));

   // reset sum
   double sumI = 0;

   std::vector <double> randx;
   randx.assign(fNvar,0.);

   // prepare maximum value
   double pmax = 0.0;

   // loop over iterations
   for(int i = 0; i <= Niter; i++)
   {
      // get random point from sampling distribution
      this -> GetRandomPointImportance(randx);

      // calculate probability at random point
      double val_f = this -> Eval(randx);

      // calculate sampling distributions at that point
      double val_g = this -> EvalSampling(randx);

      // add ratio to sum
      if (val_g > 0)
         sumI += val_f / val_g;

      // search for maximum probability
      if (val_f > pmax)
      {
         // set new maximum value
         pmax = val_f;

         // delete old best fit parameter values
         fBestFitParameters.clear();

         // write best fit parameters
         for(int i = 0; i < fNvar; i++)
            fBestFitParameters.push_back(randx.at(i));
      }

      // write intermediate results
      if((i+1)%100000 == 0)
         BCLog::Out(BCLog::debug, BCLog::debug,
            Form("BCIntegrate::IntegralImportance. Iteration %d, integral: %d.", i+1, sumI/(i+1)));
   }

   // calculate integral
   double result=sumI/Niter;

   // print debug information
   BCLog::Out(BCLog::debug, BCLog::debug, Form("BCIntegrate::IntegralImportance. Integral %f in %i iterations.", result, Niter));

   return result;
}

// *********************************************
TH1D* BCIntegrate::Marginalize(BCParameter * parameter)
{
   BCLog::Out(BCLog::detail, BCLog::detail,
                   Form(" --> Marginalizing model wrt. parameter %s using method %d.", parameter->GetName().data(), fMarginalizationMethod));

   switch(fMarginalizationMethod)
   {
      case BCIntegrate::kMargMonteCarlo:
         return MarginalizeByIntegrate(parameter);

      case BCIntegrate::kMargMetropolis:
         return MarginalizeByMetro(parameter);
   }

   BCLog::Out(BCLog::warning, BCLog::warning,
      Form("BCIntegrate::Marginalize. Invalid marginalization method: %d. Return 0.", fMarginalizationMethod));

   return 0;
}

// *********************************************
TH2D * BCIntegrate::Marginalize(BCParameter * parameter1, BCParameter * parameter2)
{
   switch(fMarginalizationMethod)
   {
      case BCIntegrate::kMargMonteCarlo:
         return MarginalizeByIntegrate(parameter1,parameter2);

      case BCIntegrate::kMargMetropolis:
         return MarginalizeByMetro(parameter1,parameter2);
   }

   BCLog::Out(BCLog::warning, BCLog::warning,
      Form("BCIntegrate::Marginalize. Invalid marginalization method: %d. Return 0.", fMarginalizationMethod));

   return 0;
}

// *********************************************
TH1D* BCIntegrate::MarginalizeByIntegrate(BCParameter * parameter)
{
   // set parameter to marginalize
   this->ResetVarlist(1);
   int index = parameter->GetIndex();
   this->UnsetVar(index);

   // define histogram
   double hmin = parameter -> GetLowerLimit();
   double hmax = parameter -> GetUpperLimit();
   double hdx  = (hmax - hmin) / double(fNbins);
   TH1D * hist = new TH1D("hist","", fNbins, hmin, hmax);

   // integrate
   std::vector <double> randx;
   randx.assign(fNvar, 0.0);

   for(int i=0;i<=fNbins;i++)
   {
      randx[index] = hmin + (double)i * hdx;

      double val = IntegralMC(randx);
      hist->Fill(randx[index], val);
   }

   // normalize
   hist -> Scale( 1./hist->Integral("width") );

   return hist;
}

// *********************************************
TH2D * BCIntegrate::MarginalizeByIntegrate(BCParameter * parameter1, BCParameter * parameter2)
{
   // set parameter to marginalize
   this->ResetVarlist(1);
   int index1 = parameter1->GetIndex();
   this->UnsetVar(index1);
   int index2 = parameter2->GetIndex();
   this->UnsetVar(index2);

   // define histogram
   double hmin1 = parameter1 -> GetLowerLimit();
   double hmax1 = parameter1 -> GetUpperLimit();
   double hdx1  = (hmax1 - hmin1) / double(fNbins);

   double hmin2 = parameter2 -> GetLowerLimit();
   double hmax2 = parameter2 -> GetUpperLimit();
   double hdx2  = (hmax2 - hmin2) / double(fNbins);

   TH2D * hist = new TH2D(Form("hist_%s_%s", parameter1 -> GetName().data(), parameter2 -> GetName().data()),"",
         fNbins, hmin1, hmax1,
         fNbins, hmin2, hmax2);

   // integrate
   std::vector <double> randx;
   randx.assign(fNvar, 0.0);

   for(int i=0;i<=fNbins;i++)
   {
      randx[index1] = hmin1 + (double)i * hdx1;
      for(int j=0;j<=fNbins;j++)
      {
         randx[index2] = hmin2 + (double)j * hdx2;

         double val = IntegralMC(randx);
         hist->Fill(randx[index1],randx[index2], val);
      }
   }

   // normalize
   hist -> Scale(1.0/hist->Integral("width"));

   return hist;
}

// *********************************************
TH1D * BCIntegrate::MarginalizeByMetro(BCParameter * parameter)
{
   int niter = fMarkovChainNIterations;

   if (fMarkovChainAutoN == true)
      niter = fNbins*fNbins*fNSamplesPer2DBin*fNvar;

   BCLog::Out(BCLog::detail, BCLog::detail,
      Form(" --> Number of samples in Metropolis marginalization: %d.", niter));

   // set parameter to marginalize
   int index = parameter->GetIndex();

   // define histogram
   double hmin = parameter -> GetLowerLimit();
   double hmax = parameter -> GetUpperLimit();
   TH1D * hist = new TH1D("hist","", fNbins, hmin, hmax);

   // prepare Metro
   std::vector <double> randx;
   randx.assign(fNvar, 0.0);
   InitMetro();

   for(int i=0;i<=niter;i++)
   {
      GetRandomPointMetro(randx);
      hist->Fill(randx[index]);
   }

   // normalize
   hist -> Scale(1.0/hist->Integral("width"));

   return hist;
}

// *********************************************
TH2D * BCIntegrate::MarginalizeByMetro(BCParameter * parameter1, BCParameter * parameter2)
{
   int niter=fNbins*fNbins*fNSamplesPer2DBin*fNvar;

   // set parameter to marginalize
   int index1 = parameter1->GetIndex();
   int index2 = parameter2->GetIndex();

   // define histogram
   double hmin1 = parameter1 -> GetLowerLimit();
   double hmax1 = parameter1 -> GetUpperLimit();

   double hmin2 = parameter2 -> GetLowerLimit();
   double hmax2 = parameter2 -> GetUpperLimit();

   TH2D * hist = new TH2D(Form("hist_%s_%s", parameter1 -> GetName().data(), parameter2 -> GetName().data()),"",
         fNbins, hmin1, hmax1,
         fNbins, hmin2, hmax2);

   // prepare Metro
   std::vector <double> randx;
   randx.assign(fNvar, 0.0);
   InitMetro();

   for(int i=0;i<=niter;i++)
   {
      GetRandomPointMetro(randx);
      hist->Fill(randx[index1],randx[index2]);
   }

   // normalize
   hist -> Scale(1.0/hist->Integral("width"));

   return hist;
}

// *********************************************
int BCIntegrate::MarginalizeAllByMetro(const char * name="")
{
   int niter=fNbins*fNbins*fNSamplesPer2DBin*fNvar;

   BCLog::Out(BCLog::detail, BCLog::detail,
         Form(" --> Number of samples in Metropolis marginalization: %d.", niter));

   // define 1D histograms
   for(int i=0;i<fNvar;i++)
   {
      double hmin1 = fx->at(i) -> GetLowerLimit();
      double hmax1 = fx->at(i) -> GetUpperLimit();

      TH1D * h1 = new TH1D(
            TString::Format("h%s_%s", name, fx->at(i) -> GetName().data()),"",
            fNbins, hmin1, hmax1);

      fHProb1D.push_back(h1);
   }

   // define 2D histograms
   for(int i=0;i<fNvar-1;i++)
      for(int j=i+1;j<fNvar;j++)
      {
         double hmin1 = fx->at(i) -> GetLowerLimit();
         double hmax1 = fx->at(i) -> GetUpperLimit();

         double hmin2 = fx->at(j) -> GetLowerLimit();
         double hmax2 = fx->at(j) -> GetUpperLimit();

         TH2D * h2 = new TH2D(
            TString::Format("h%s_%s_%s", name, fx->at(i) -> GetName().data(), fx->at(j) -> GetName().data()),"",
            fNbins, hmin1, hmax1,
            fNbins, hmin2, hmax2);

         fHProb2D.push_back(h2);
      }

   // get number of 2d distributions
   int nh2d = fHProb2D.size();

   BCLog::Out(BCLog::detail, BCLog::detail,
         Form(" --> Marginalizing %d 1D distributions and %d 2D distributions.", fNvar, nh2d));

   // prepare function fitting
   double dx = 0.0;
   double dy = 0.0;

   if (fFitFunctionIndexX >= 0)
   {
      dx = (fDataPointUpperBoundaries -> GetValue(fFitFunctionIndexX) -
            fDataPointLowerBoundaries -> GetValue(fFitFunctionIndexX))
            / double(fErrorBandNbinsX);

      dx = (fDataPointUpperBoundaries -> GetValue(fFitFunctionIndexY) -
            fDataPointLowerBoundaries -> GetValue(fFitFunctionIndexY))
            / double(fErrorBandNbinsY);

      fErrorBandXY = new TH2D(
            TString::Format("errorbandxy_%d",BCLog::GetHIndex()), "",
            fErrorBandNbinsX,
            fDataPointLowerBoundaries -> GetValue(fFitFunctionIndexX) - 0.5 * dx,
            fDataPointUpperBoundaries -> GetValue(fFitFunctionIndexX) + 0.5 * dx,
            fErrorBandNbinsY,
            fDataPointLowerBoundaries -> GetValue(fFitFunctionIndexY) - 0.5 * dy,
            fDataPointUpperBoundaries -> GetValue(fFitFunctionIndexY) + 0.5 * dy);
      fErrorBandXY -> SetStats(kFALSE);

      for (int ix = 1; ix <= fErrorBandNbinsX; ++ix)
         for (int iy = 1; iy <= fErrorBandNbinsX; ++iy)
            fErrorBandXY -> SetBinContent(ix, iy, 0.0);
   }

   // prepare Metro
   std::vector <double> randx;

   randx.assign(fNvar, 0.0);
   InitMetro();

   // reset counter
   fNacceptedMCMC = 0;

   // run Metro and fill histograms
   for(int i=0;i<=niter;i++)
   {
      GetRandomPointMetro(randx);

      // save this point to the markov chain in the ROOT file
      if (fFlagWriteMarkovChain)
         {
            fMCMCIteration = i;
            fMarkovChainTree -> Fill();
         }

      for(int j=0;j<fNvar;j++)
         fHProb1D[j] -> Fill( randx[j] );

      int ih=0;
      for(int j=0;j<fNvar-1;j++)
         for(int k=j+1;k<fNvar;k++)
         {
            fHProb2D[ih] -> Fill(randx[j],randx[k]);
            ih++;
         }

      if((i+1)%100000==0)
         BCLog::Out(BCLog::debug, BCLog::debug,
            Form("BCIntegrate::MarginalizeAllByMetro. %d samples done.",i+1));

      // function fitting

      if (fFitFunctionIndexX >= 0)
      {
         // loop over all possible x values ...
         if (fErrorBandContinuous)
         {
            double x = 0;

            for (int ix = 0; ix < fErrorBandNbinsX; ix++)
            {
               // calculate x
               x = fErrorBandXY -> GetXaxis() -> GetBinCenter(ix + 1);

               // calculate y
               std::vector <double> xvec;
               xvec.push_back(x);
               double y = this -> FitFunction(xvec, randx);
               xvec.clear();

               // fill histogram
               fErrorBandXY -> Fill(x, y);
            }
         }

         // ... or evaluate at the data point x-values
         else
         {
            int ndatapoints = int(fErrorBandX.size());
            double x = 0;

            for (int ix = 0; ix < ndatapoints; ++ix)
            {
               // calculate x
               x = fErrorBandX.at(ix);

               // calculate y
               std::vector <double> xvec;
               xvec.push_back(x);
               double y = this -> FitFunction(xvec, randx);
               xvec.clear();

               // fill histogram
               fErrorBandXY -> Fill(x, y);
            }
         }
      }
   }

   // normalize histograms
   for(int i=0;i<fNvar;i++)
      fHProb1D[i] -> Scale( 1./fHProb1D[i]->Integral("width") );

   for (int i=0;i<nh2d;i++)
      fHProb2D[i] -> Scale( 1./fHProb2D[i]->Integral("width") );

   if (fFitFunctionIndexX >= 0)
      fErrorBandXY -> Scale(1.0/fErrorBandXY -> Integral() * fErrorBandXY -> GetNbinsX());

   if (fFitFunctionIndexX >= 0)
   {
      for (int ix = 1; ix <= fErrorBandNbinsX; ix++)
      {
         double sum = 0;

         for (int iy = 1; iy <= fErrorBandNbinsY; iy++)
            sum += fErrorBandXY -> GetBinContent(ix, iy);

         for (int iy = 1; iy <= fErrorBandNbinsY; iy++)
         {
            double newvalue = fErrorBandXY -> GetBinContent(ix, iy) / sum;
            fErrorBandXY -> SetBinContent(ix, iy, newvalue);
         }
      }
   }

   BCLog::Out(BCLog::detail, BCLog::detail,
         Form("BCIntegrate::MarginalizeAllByMetro done with %i trials and %i accepted trials. Efficiency is %f",fNmetro, fNacceptedMCMC, double(fNacceptedMCMC)/double(fNmetro)));

   return fNvar+nh2d;
}

// *********************************************
TH1D * BCIntegrate::GetH1D(int parIndex)
{
   if(fHProb1D.size()==0)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         "BCModel::GetH1D. MarginalizeAll() has to be run prior to this to fill the distributions.");
      return 0;
   }

   if(parIndex<0 || parIndex>fNvar)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         Form("BCIntegrate::GetH1D. Parameter index %d is invalid.",parIndex));
      return 0;
   }

   return fHProb1D[parIndex];
}

// *********************************************
int BCIntegrate::GetH2DIndex(int parIndex1, int parIndex2)
{
   if(parIndex1>fNvar-1 || parIndex1<0)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         Form("BCIntegrate::GetH2DIndex. Parameter index %d is invalid", parIndex1));
      return -1;
   }

   if(parIndex2>fNvar-1 || parIndex2<0)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         Form("BCIntegrate::GetH2DIndex. Parameter index %d is invalid", parIndex2));
      return -1;
   }

   if(parIndex1==parIndex2)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         Form("BCIntegrate::GetH2DIndex. Parameters have equal indeces: %d , %d", parIndex1, parIndex2));
      return -1;
   }

   if(parIndex1>parIndex2)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         "BCIntegrate::GetH2DIndex. First parameters must be smaller than second (sorry).");
      return -1;
   }

   int index=0;
   for(int i=0;i<fNvar-1;i++)
      for(int j=i+1;j<fNvar;j++)
      {
         if(i==parIndex1 && j==parIndex2)
            return index;
         index++;
      }

   BCLog::Out(BCLog::warning, BCLog::warning,
      "BCIntegrate::GetH2DIndex. Invalid index combination.");

   return -1;
}

// *********************************************
TH2D * BCIntegrate::GetH2D(int parIndex1, int parIndex2)
{
   if(fHProb2D.size()==0)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         "BCModel::GetH2D. MarginalizeAll() has to be run prior to this to fill the distributions.");
      return 0;
   }

   int hindex = this -> GetH2DIndex(parIndex1, parIndex2);
   if(hindex==-1)
      return 0;

   if(hindex>(int)fHProb2D.size()-1)
   {
      BCLog::Out(BCLog::warning, BCLog::warning,
         "BCIntegrate::GetH2D. Got invalid index from GetH2DIndex(). Something went wrong.");
      return 0;
   }

   return fHProb2D[hindex];
}

// *********************************************
double BCIntegrate::GetRandomPoint(std::vector <double> &x)
{
   this->GetRandomVector(x);

   for(int i=0;i<fNvar;i++)
      x[i]=fMin[i]+x[i]*(fMax[i]-fMin[i]);

   return this->Eval(x);
}

// *********************************************
double BCIntegrate::GetRandomPointImportance(std::vector <double> &x)
{
   double p = 1.1;
   double g = 1.0;

   while (p>g)
   {
      this->GetRandomVector(x);

      for(int i=0;i<fNvar;i++)
         x[i] = fMin[i] + x[i] * (fMax[i]-fMin[i]);

      p = fRandom->Rndm();

      g = this -> EvalSampling(x);
   }

   return this->Eval(x);
}

// *********************************************
void BCIntegrate::InitMetro()
{
   fNmetro=0;

   if (fXmetro0.size() <= 0)
   {
      // start in the center of the phase space
      for(int i=0;i<fNvar;i++)
         fXmetro0.push_back((fMin[i]+fMax[i])/2.0);
   }

   fMarkovChainValue = this ->  LogEval(fXmetro0);

   // run metropolis for a few times and dump the result... (to forget the initial position)
   std::vector <double> x;
   x.assign(fNvar,0.);

   for(int i=0;i<1000;i++)
      GetRandomPointMetro(x);

   fNmetro = 0;
}

// *********************************************
void BCIntegrate::GetRandomPointMetro(std::vector <double> &x)
{
   // get new point
   this->GetRandomVectorMetro(fXmetro1);

   // scale the point to the allowed region and stepsize
   int in=1;
   for(int i=0;i<fNvar;i++)
   {
      fXmetro1[i] = fXmetro0[i] + fXmetro1[i] * (fMax[i]-fMin[i]);

      // check whether the generated point is inside the allowed region
      if( fXmetro1[i]<fMin[i] || fXmetro1[i]>fMax[i] )
         in=0;
   }

   // calculate the log probabilities and compare old and new point

   double p0 = fMarkovChainValue; // old point
   double p1 = 0; // new point
   int accept=0;

   if(in)
   {
      p1 = this -> LogEval(fXmetro1);

      if(p1>=p0)
         accept=1;
      else
      {
         double r=log(fRandom->Rndm());
         if(r<p1-p0)
            accept=1;
      }
   }

   // fill the return point after the decision
   if(accept)
   {
      // increase counter
      fNacceptedMCMC++;

      for(int i=0;i<fNvar;i++)
      {
         fXmetro0[i]=fXmetro1[i];
         x[i]=fXmetro1[i];
         fMarkovChainValue = p1;
      }
   }
   else
      for(int i=0;i<fNvar;i++)
      {
         x[i]=fXmetro0[i];
         fXmetro1[i] = x[i];
         fMarkovChainValue = p0;
      }

   fNmetro++;
}

// *********************************************
void BCIntegrate::GetRandomPointSamplingMetro(std::vector <double> &x)
{
   // get new point
   this->GetRandomVectorMetro(fXmetro1);

   // scale the point to the allowed region and stepsize
   int in=1;
   for(int i=0;i<fNvar;i++)
   {
      fXmetro1[i] = fXmetro0[i] + fXmetro1[i] * (fMax[i]-fMin[i]);

      // check whether the generated point is inside the allowed region
      if( fXmetro1[i]<fMin[i] || fXmetro1[i]>fMax[i] )
         in=0;
   }

   // calculate the log probabilities and compare old and new point
   double p0 = this -> LogEvalSampling(fXmetro0); // old point
   double p1=0; // new point
   if(in)
      p1= this -> LogEvalSampling(fXmetro1);

   // compare log probabilities
   int accept=0;
   if(in)
   {
      if(p1>=p0)
         accept=1;
      else
      {
         double r=log(fRandom->Rndm());
         if(r<p1-p0)
            accept=1;
      }
   }

   // fill the return point after the decision
   if(accept)
      for(int i=0;i<fNvar;i++)
      {
         fXmetro0[i]=fXmetro1[i];
         x[i]=fXmetro1[i];
      }
   else
      for(int i=0;i<fNvar;i++)
         x[i]=fXmetro0[i];

   fNmetro++;
}

// *********************************************
void BCIntegrate::GetRandomVector(std::vector <double> &x)
{
   double * randx = new double[fNvar];

   fRandom -> RndmArray(fNvar, randx);

   for(int i=0;i<fNvar;i++)
      x[i] = randx[i];

   delete[] randx;
   randx = 0;
}

// *********************************************
void BCIntegrate::GetRandomVectorMetro(std::vector <double> &x)
{
   double * randx = new double[fNvar];

   fRandom -> RndmArray(fNvar, randx);

   for(int i=0;i<fNvar;i++)
      x[i] = 2.0 * (randx[i] - 0.5) * fMarkovChainStepSize;

   delete[] randx;
   randx = 0;
}

// *********************************************
TMinuit * BCIntegrate::GetMinuit()
{
   if (!fMinuit)
      fMinuit = new TMinuit();

   return fMinuit;
}

// *********************************************

void BCIntegrate::FindModeMinuit(std::vector<double> start, int printlevel)
{
   bool have_start = true;

   // check start values
   if (int(start.size()) != fNvar)
      have_start = false;

   // set global this
   global_this = this;

   // define minuit
   if (fMinuit)
      delete fMinuit;
   fMinuit = new TMinuit(fNvar);

   // set print level
   fMinuit -> SetPrintLevel(printlevel);

   // set function
   fMinuit -> SetFCN(&BCIntegrate::FCNLikelihood);

   // set UP for likelihood 
   fMinuit -> SetErrorDef(0.5); 

   // set parameters
   int flag;
   for (int i = 0; i < fNvar; i++)
   {
      double starting_point = (fMin[i]+fMax[i])/2.;
      if(have_start)
         starting_point = start[i];
      fMinuit -> mnparm(i,
                                 fx -> at(i) -> GetName().data(),
                                 starting_point,
                                 (fMax[i]-fMin[i])/100.0,
                                 fMin[i],
                                 fMax[i],
                                 flag);
   }

   // do mcmc minimization
   // fMinuit -> mnseek(); 

   // do minimization
   fMinuit -> mnexcm("MIGRAD", fMinuitArglist, 2, flag);

   // improve search for local minimum 
   // fMinuit -> mnimpr(); 

   // copy flag
   fMinuitErrorFlag = flag;

   // check if mode found by minuit is better than previous estimation 
   double probmax = 0; 
   bool valid = false; 
   
   if ( int(fBestFitParameters.size()) == fNvar) 
      {
         valid = true; 
         for (int i = 0; i < fNvar; ++i)
            if (fBestFitParameters.at(i) < fMin[i] || fBestFitParameters.at(i) > fMax[i]) 
               valid= false; 

         if (valid)
            probmax = this -> Eval( fBestFitParameters ); 
      }

   std::vector<double> tempvec; 
   for (int i = 0; i < fNvar; i++)
      {
         double par;
         double parerr; 
         fMinuit -> GetParameter(i, par, parerr);
         tempvec.push_back(par); 
      }
   double probmaxminuit = this -> Eval( tempvec ); 
   
   // set best fit parameters
   if ( (probmaxminuit > probmax && !fFlagIgnorePrevOptimization) || !valid || fFlagIgnorePrevOptimization) 
      {
         fBestFitParameters.clear();
         fBestFitParameterErrors.clear();

         for (int i = 0; i < fNvar; i++)
            {
               double par;
               double parerr;
               fMinuit -> GetParameter(i, par, parerr);
               fBestFitParameters.push_back(par);
               fBestFitParameterErrors.push_back(parerr);
            }

         // set optimization method used to find the mode 
         fOptimizationMethodMode = BCIntegrate::kOptMinuit; 
      }

   // delete minuit
   delete fMinuit;
   fMinuit = 0;

   return;
}

// *********************************************

void BCIntegrate::FindModeSA(std::vector<double> start)
{
   // note: if f(x) is the function to be minimized, then
   // f(x) := - this->LogEval(parameters)

   bool have_start = true;
   std::vector<double> x, y, best_fit; // vectors for current state, new proposed state and best fit up to now
   double fval_x, fval_y, fval_best_fit; // function values at points x, y and best_fit (we save them rather than to re-calculate them every time)
   int t = 1; // time iterator

   // check start values
   if (int(start.size()) != fNvar)
      have_start = false;

   // if no starting point is given, set to center of parameter space
   if ( !have_start )
   {
      start.clear();
      for (int i = 0; i < fNvar; i++)
         start.push_back((fMin[i]+fMax[i])/2.);
   }

   // set current state and best fit to starting point
   x.clear();
   best_fit.clear();
   for (int i = 0; i < fNvar; i++)
   {
      x.push_back(start[i]);
      best_fit.push_back(start[i]);
   }
   // calculate function value at starting point
   fval_x = fval_best_fit = this->LogEval(x);

   // run while still "hot enough"
   while ( this->SATemperature(t) > fSATmin )
   {
      // generate new state
      y = this->GetProposalPointSA(x, t);

      // check if the proposed point is inside the phase space
      // if not, reject it
      bool is_in_ranges = true;

      for (int i = 0; i < fNvar; i++)
         if (y[i] > fMax[i] || y[i] < fMin[i])
            is_in_ranges = false;
      
      if ( !is_in_ranges )
         ; // do nothing...
      else
      {
         // calculate function value at new point
         fval_y = this->LogEval(y);

         // is it better than the last one?
         // if so, update state and chef if it is the new best fit...
         if (fval_y >= fval_x)
         {
            x.clear();
            for (int i = 0; i < fNvar; i++)
               x.push_back(y[i]);

            fval_x = fval_y;

            if (fval_y > fval_best_fit)
            {
               best_fit.clear();
               for (int i = 0; i < fNvar; i++)
                  best_fit.push_back(y[i]);
               
               fval_best_fit = fval_y;
            }
         }
         // ...else, only accept new state w/ certain probability
         else
         {
            if (fRandom->Rndm() <= exp( (fval_y - fval_x)
                  / this->SATemperature(t) ))
            {
               x.clear();
               for (int i = 0; i < fNvar; i++)
                  x.push_back(y[i]);
               
               fval_x = fval_y;
            }
         }
      }

      // update tree variables 
      fSANIterations = t; 
      fSATemperature = this -> SATemperature(t); 
      fSALogProb = fval_x; 
      fSAx.clear(); 
      for (int i = 0; i < fNvar; ++i)
         fSAx.push_back(x[i]); 

      // fill tree
      if (fFlagWriteSAToFile)
         fTreeSA -> Fill(); 

      // increate t 
      t++;
   }

   // check if mode found by minuit is better than previous estimation 
   double probmax = 0; 
   bool valid = false; 
   
   if ( int(fBestFitParameters.size()) == fNvar) 
      {
         valid = true; 
         for (int i = 0; i < fNvar; ++i)
            if (fBestFitParameters.at(i) < fMin[i] || fBestFitParameters.at(i) > fMax[i]) 
               valid= false; 

         if (valid)
            probmax = this -> Eval( fBestFitParameters ); 
      }

   double probmaxsa = this -> Eval( best_fit); 

   if ( (probmaxsa > probmax  && !fFlagIgnorePrevOptimization) || !valid || fFlagIgnorePrevOptimization) 
      {
         // set best fit parameters
         fBestFitParameters.clear();
         fBestFitParameterErrors.clear(); 

         for (int i = 0; i < fNvar; i++)
            {
               fBestFitParameters.push_back(best_fit[i]);
               fBestFitParameterErrors.push_back(0.0); 
            }

         // set optimization moethod used to find the mode 
         fOptimizationMethodMode = BCIntegrate::kOptSA;        
      }

   return;
}

// *********************************************

double BCIntegrate::SATemperature(double t)
{
   // do we have Cauchy (default), Boltzmann or custom annealing schedule?
   if (this->fSASchedule == BCIntegrate::kSABoltzmann)
      return this->SATemperatureBoltzmann(t);
   else if (this->fSASchedule == BCIntegrate::kSACauchy)
      return this->SATemperatureCauchy(t);
   else
      return this->SATemperatureCustom(t);
}

// *********************************************

double BCIntegrate::SATemperatureBoltzmann(double t)
{
   return fSAT0 / log((double)(t + 1));
}

// *********************************************

double BCIntegrate::SATemperatureCauchy(double t)
{
   return fSAT0 / (double)t;
}

// *********************************************

double BCIntegrate::SATemperatureCustom(double t)
{
   BCLog::OutError("BCIntegrate::SATemperatureCustom : No custom temperature schedule defined");
   return 0.;
}

// *********************************************

std::vector<double> BCIntegrate::GetProposalPointSA(std::vector<double> x, int t)
{
   // do we have Cauchy (default), Boltzmann or custom annealing schedule?
   if (this->fSASchedule == BCIntegrate::kSABoltzmann)
      return this->GetProposalPointSABoltzmann(x, t);
   else if (this->fSASchedule == BCIntegrate::kSACauchy)
      return this->GetProposalPointSACauchy(x, t);
   else
      return this->GetProposalPointSACustom(x, t);
}

// *********************************************

std::vector<double> BCIntegrate::GetProposalPointSABoltzmann(std::vector<double> x, int t)
{
   std::vector<double> y;
   y.clear();
   double new_val, norm;

   for (int i = 0; i < fNvar; i++)
   {
      norm = (fMax[i] - fMin[i]) * this->SATemperature(t) / 2.;
      new_val = x[i] + norm * fRandom->Gaus();
      y.push_back(new_val);
   }

   return y;
}

// *********************************************

std::vector<double> BCIntegrate::GetProposalPointSACauchy(std::vector<double> x, int t)
{
   std::vector<double> y;
   y.clear();

   if (fNvar == 1)
   {
      double cauchy, new_val, norm;

      norm = (fMax[0] - fMin[0]) * this->SATemperature(t) / 2.;
      cauchy = tan(3.14159 * (fRandom->Rndm() - 0.5));
      new_val = x[0] + norm * cauchy;
      y.push_back(new_val);
   }
   else
   {
      // use sampling to get radial n-dim Cauchy distribution

      // first generate a random point uniformly distributed on a
      // fNvar-dimensional hypersphere
      y = this->SAHelperGetRandomPointOnHypersphere();

      // scale the vector by a random factor determined by the radial
      // part of the fNvar-dimensional Cauchy distribution
      double radial = this->SATemperature(t) * this->SAHelperGetRadialCauchy();

      // scale y by radial part and the size of dimension i in phase space
      // afterwards, move by x
      for (int i = 0; i < fNvar; i++)
         y[i] = (fMax[i] - fMin[i]) * y[i] * radial / 2. + x[i];
   }

   return y;
}

// *********************************************

std::vector<double> BCIntegrate::GetProposalPointSACustom(std::vector<double> x, int t)
{
   BCLog::OutError("BCIntegrate::GetProposalPointSACustom : No custom proposal function defined");
   return std::vector<double>(fNvar);
}

// *********************************************

std::vector<double> BCIntegrate::SAHelperGetRandomPointOnHypersphere()
{
   std::vector<double> rand_point(fNvar);
   
   // This method can only be called with fNvar >= 2 since the 1-dim case
   // is already hard wired into the Cauchy annealing proposal function.
   // To speed things up, hard-code fast method for 2 and dimensions.
   // The algorithm for 2D can be found at
   // http://mathworld.wolfram.com/CirclePointPicking.html
   // For 3D just using ROOT's algorithm.
   if (fNvar == 2)
   {
      double x1, x2, s;
      do {
         x1 = fRandom->Rndm() * 2. - 1.;
         x2 = fRandom->Rndm() * 2. - 1.;
         s = x1*x1 + x2*x2;
      } while (s >= 1);
      
      rand_point[0] = (x1*x1 - x2*x2) / s;
      rand_point[1] = (2.*x1*x2) / s;
   }
   else if (fNvar == 3)
   {
      fRandom->Sphere(rand_point[0], rand_point[1], rand_point[2], 1.0);
   }
   else
   {
      double s = 0.,
         gauss_num;

      for (int i = 0; i < fNvar; i++)
      {
         gauss_num = fRandom->Gaus();
         rand_point[i] = gauss_num;
         s += gauss_num * gauss_num;
      }
      s = sqrt(s);

      for (int i = 0; i < fNvar; i++)
         rand_point[i] = rand_point[i] / s;
   }
   
   return rand_point;
}

// *********************************************

double BCIntegrate::SAHelperGetRadialCauchy()
{
   // theta is sampled from a rather complicated distribution,
   // so first we create a lookup table with 10000 random numbers
   // once and then, each time we need a new random number,
   // we just look it up in the table.
   double theta;

   // static vectors for theta-sampling-map
   static double map_u[10001];
   static double map_theta[10001];
   static bool initialized = false;
   static int map_dimension = 0;

   // is the lookup-table already initialized? if not, do it!
   if (!initialized || map_dimension != fNvar)
   {
      double init_theta;
      double beta = this->SAHelperSinusToNIntegral(fNvar - 1, 1.57079632679);
      
      for (int i = 0; i <= 10000; i++)
      {
         init_theta = 3.14159265 * (double)i / 5000.;
         map_theta[i] = init_theta;

         map_u[i] = this->SAHelperSinusToNIntegral(fNvar - 1, init_theta) / beta;
      }

      map_dimension = fNvar;
      initialized = true;
   } // initializing is done.

   // generate uniform random number for sampling
   double u = fRandom->Rndm();

   // Find the two elements just greater than and less than u
   // using a binary search (O(log(N))).
   int lo = 0;
   int up = 10000;
   int mid;

   while (up != lo)
   {
      mid = ((up - lo + 1) / 2) + lo;

      if (u >= map_u[mid])
         lo = mid;
      else
         up = mid - 1;
   }
   up++;

   // perform linear interpolation:
   theta = map_theta[lo] + (u - map_u[lo]) / (map_u[up] - map_u[lo])
      * (map_theta[up] - map_theta[lo]);

   return tan(theta);
}

// *********************************************

double BCIntegrate::SAHelperSinusToNIntegral(int dim, double theta)
{
   if (dim < 1)
      return theta;
   else if (dim == 1)
      return (1. - cos(theta));
   else if (dim == 2)
      return 0.5 * (theta - sin(theta) * cos(theta));
   else if (dim == 3)
      return (2. - sin(theta) * sin(theta) * cos(theta) - 2. * cos(theta)) / 3.;
   else
      return - pow(sin(theta), (double)(dim - 1)) * cos(theta) / (double)dim
         + (double)(dim - 1) / (double)dim
         * this->SAHelperSinusToNIntegral(dim - 2, theta);
}

// *********************************************

void BCIntegrate::SetMode(std::vector <double> mode)
{
   if((int)mode.size() == fNvar)
   {
      fBestFitParameters.clear();
      for (int i = 0; i < fNvar; i++)
         fBestFitParameters.push_back(mode[i]);
   }
}

// *********************************************

void BCIntegrate::FCNLikelihood(int &npar, double * grad, double &fval, double * par, int flag)
{
   // copy parameters
   std::vector <double> parameters;

   int n = global_this -> GetNvar();

   for (int i = 0; i < n; i++)
      parameters.push_back(par[i]);

   fval = - global_this -> LogEval(parameters);

   // delete parameters
   parameters.clear();
}

// *********************************************

//void BCIntegrate::FindModeMCMC(int flag_run)
void BCIntegrate::FindModeMCMC()
{
   // call PreRun
// if (flag_run == 0)
// if (!fMCMCFlagPreRun)
//    this -> MCMCMetropolisPreRun();

   // find global maximum
   // double probmax = (this -> MCMCGetMaximumLogProb()).at(0);
   
   double probmax = 0; 

   if ( int(fBestFitParameters.size()) == fNvar)
      {
         probmax = this -> Eval( fBestFitParameters ); 
         //       fBestFitParameters = this -> MCMCGetMaximumPoint(0);
      }

   // loop over all chains and find the maximum point
   for (int i = 0; i < fMCMCNChains; ++i)
   {
      double prob = exp( (this -> MCMCGetMaximumLogProb()).at(i));

      // copy the point into the vector
      if ( (prob >= probmax && !fFlagIgnorePrevOptimization) || fFlagIgnorePrevOptimization)
      {
         probmax = prob;

         fBestFitParameters.clear();
         fBestFitParameterErrors.clear();
         fBestFitParameters = this -> MCMCGetMaximumPoint(i);

         for (int j = 0; j < fNvar; ++j)
            fBestFitParameterErrors.push_back(0.0); 

         fOptimizationMethodMode = BCIntegrate::kOptMetropolis; 
      }
   }

}

// *********************************************
void BCIntegrate::CubaIntegrand(const int *ndim, const double xx[],
      const int *ncomp, double ff[])
{
#ifdef HAVE_CUBA_H
   // scale variables
   double jacobian = 1.0;

   std::vector<double> scaled_parameters;

   for (int i = 0; i < *ndim; i++)
   {
      double range = global_this -> fx -> at(i) -> GetUpperLimit() -  global_this -> fx -> at(i) -> GetLowerLimit();

      // multiply range to jacobian
      jacobian *= range;

      // get the scaled parameter value
      scaled_parameters.push_back(global_this -> fx -> at(i) -> GetLowerLimit() + xx[i] * range);
   }

   // call function to integrate
   ff[0] = global_this -> Eval(scaled_parameters);

   // multiply jacobian
   ff[0] *= jacobian;

   // multiply fudge factor
   ff[0] *= 1e99;

   // remove parameter vector
   scaled_parameters.clear();
#else
   BCLog::Out(BCLog::error,BCLog::error,"!!! This version of BAT is compiled without Cuba.");
   BCLog::Out(BCLog::error,BCLog::error,"    Use other integration methods or install Cuba and recompile BAT.");
#endif
}

// *********************************************
double BCIntegrate::CubaIntegrate()
{
#ifdef HAVE_CUBA_H
   double EPSREL = 1e-3;
   double EPSABS = 1e-12;
   double VERBOSE   = 0;
   double MINEVAL   = 0;
   double MAXEVAL   = 2000000;
   double NSTART    = 25000;
   double NINCREASE = 25000;

   std::vector<double> parameters_double;
   std::vector<double>    parameters_int;

   parameters_double.push_back(EPSREL);
   parameters_double.push_back(EPSABS);

   parameters_int.push_back(VERBOSE);
   parameters_int.push_back(MINEVAL);
   parameters_int.push_back(MAXEVAL);
   parameters_int.push_back(NSTART);
   parameters_int.push_back(NINCREASE);

   // print to log
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Running Cuba/Vegas integation over %i dimensions.", fNvar));
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Maximum number of iterations: %i", (int)MAXEVAL));
   BCLog::Out(BCLog::detail, BCLog::detail, Form(" --> Aimed relative precision:     %e", EPSREL));

   return this -> CubaIntegrate(0, parameters_double, parameters_int);
#else
   BCLog::Out(BCLog::error,BCLog::error,"!!! This version of BAT is compiled without Cuba.");
   BCLog::Out(BCLog::error,BCLog::error,"    Use other integration methods or install Cuba and recompile BAT.");
   return 0.;
#endif
}

// *********************************************
double BCIntegrate::CubaIntegrate(int method, std::vector<double> parameters_double, std::vector<double> parameters_int)
{
#ifdef HAVE_CUBA_H
   const int NDIM      = int(fx ->size());
   const int NCOMP     = 1;

   const double EPSREL = parameters_double[0];
   const double EPSABS = parameters_double[1];
   const int VERBOSE   = int(parameters_int[0]);
   const int MINEVAL   = int(parameters_int[1]);
   const int MAXEVAL   = int(parameters_int[2]);

   int neval;
   int fail;
   int nregions;
   double integral[NCOMP];
   double error[NCOMP];
   double prob[NCOMP];

   global_this = this;

   integrand_t an_integrand = &BCIntegrate::CubaIntegrand;

   if (method == 0)
   {
      // set VEGAS specific parameters
      const int NSTART    = int(parameters_int[3]);
      const int NINCREASE = int(parameters_int[4]);

      // call VEGAS integration method
      Vegas(NDIM, NCOMP, an_integrand,
         EPSREL, EPSABS, VERBOSE, MINEVAL, MAXEVAL,
         NSTART, NINCREASE,
         &neval, &fail, integral, error, prob);
   }
   else if (method == 1)
   {
      // set SUAVE specific parameters
      const int LAST     = int(parameters_int[3]);
      const int NNEW     = int(parameters_int[4]);
      const int FLATNESS = int(parameters_int[5]);

      // call SUAVE integration method
      Suave(NDIM, NCOMP, an_integrand,
         EPSREL, EPSABS, VERBOSE | LAST, MINEVAL, MAXEVAL,
         NNEW, FLATNESS,
         &nregions, &neval, &fail, integral, error, prob);
   }
   else if (method == 2)
   {
      // set DIVONNE specific parameters
      const int KEY1         = int(parameters_int[3]);
      const int KEY2         = int(parameters_int[4]);
      const int KEY3         = int(parameters_int[5]);
      const int MAXPASS      = int(parameters_int[6]);
      const int BORDER       = int(parameters_int[7]);
      const int MAXCHISQ     = int(parameters_int[8]);
      const int MINDEVIATION = int(parameters_int[9]);
      const int NGIVEN       = int(parameters_int[10]);
      const int LDXGIVEN     = int(parameters_int[11]);
      const int NEXTRA       = int(parameters_int[12]);

      // call DIVONNE integration method
      Divonne(NDIM, NCOMP, an_integrand,
         EPSREL, EPSABS, VERBOSE, MINEVAL, MAXEVAL,
         KEY1, KEY2, KEY3, MAXPASS, BORDER, MAXCHISQ, MINDEVIATION,
         NGIVEN, LDXGIVEN, NULL, NEXTRA, NULL,
         &nregions, &neval, &fail, integral, error, prob);
   }
   else if (method == 3)
   {
      // set CUHRE specific parameters
      const int LAST = int(parameters_int[3]);
      const int KEY  = int(parameters_int[4]);

      // call CUHRE integration method
      Cuhre(NDIM, NCOMP, an_integrand,
         EPSREL, EPSABS, VERBOSE | LAST, MINEVAL, MAXEVAL, KEY,
         &nregions, &neval, &fail, integral, error, prob);
   }
   else
   {
      std::cout << " Integration method not available. " << std::endl;
      integral[0] = -1e99;
   }

   if (fail != 0)
   {
      std::cout << " Warning, integral did not converge with the given set of parameters. "<< std::endl;
      std::cout << " neval    = " << neval       << std::endl;
      std::cout << " fail     = " << fail        << std::endl;
      std::cout << " integral = " << integral[0] << std::endl;
      std::cout << " error    = " << error[0]    << std::endl;
      std::cout << " prob     = " << prob[0]     << std::endl;
   }

   return integral[0] / 1e99;
#else
   BCLog::Out(BCLog::error,BCLog::error,"!!! This version of BAT is compiled without Cuba.");
   BCLog::Out(BCLog::error,BCLog::error,"    Use other integration methods or install Cuba and recompile BAT.");
   return 0.;
#endif
}

// *********************************************
void BCIntegrate::MCMCIterationInterface()
{
   // what's within this method will be executed
   // for every iteration of the MCMC

   // fill error band
   this -> MCMCFillErrorBand();

   // do user defined stuff
   this -> MCMCUserIterationInterface();
}

// *********************************************
void BCIntegrate::MCMCFillErrorBand()
{
   if (!fFillErrorBand)
      return;

   // function fitting
   if (fFitFunctionIndexX < 0)
      return;

   // loop over all possible x values ...
   if (fErrorBandContinuous)
   {
      double x = 0;
      for (int ix = 0; ix < fErrorBandNbinsX; ix++)
      {
         // calculate x
         x = fErrorBandXY -> GetXaxis() -> GetBinCenter(ix + 1);

         // calculate y
         std::vector <double> xvec;
         xvec.push_back(x);

         // loop over all chains
         for (int ichain = 0; ichain < this -> MCMCGetNChains(); ++ichain)
         {
            // calculate y
            double y = this -> FitFunction(xvec, this -> MCMCGetx(ichain));

            // fill histogram
            fErrorBandXY -> Fill(x, y);
         }

         xvec.clear();
      }
   }
   // ... or evaluate at the data point x-values
   else
   {
      int ndatapoints = int(fErrorBandX.size());
      double x = 0;

      for (int ix = 0; ix < ndatapoints; ++ix)
      {
         // calculate x
         x = fErrorBandX.at(ix);

         // calculate y
         std::vector <double> xvec;
         xvec.push_back(x);

         // loop over all chains
         for (int ichain = 0; ichain < this -> MCMCGetNChains(); ++ichain)
         {
            // calculate y
            double y = this -> FitFunction(xvec, this -> MCMCGetx(ichain));

            // fill histogram
            fErrorBandXY -> Fill(x, y);
         }

         xvec.clear();
      }
   }
}

// *********************************************
void BCIntegrate::SAInitialize()
{
   fSAx.clear(); 
   fSAx.assign(fNvar, 0.0); 
}

// *********************************************

---