rs302_JeffreysPriorDemo.C File Reference

Detailed Description

tutorial demonstrating and validates the RooJeffreysPrior class

Jeffreys's prior is an 'objective prior' based on formal rules. It is calculated from the Fisher information matrix.

Read more: http://en.wikipedia.org/wiki/Jeffreys_prior

The analytic form is not known for most PDFs, but it is for simple cases like the Poisson mean, Gaussian mean, Gaussian sigma.

This class uses numerical tricks to calculate the Fisher Information Matrix efficiently. In particular, it takes advantage of a property of the 'Asimov data' as described in Asymptotic formulae for likelihood-based tests of new physics Glen Cowan, Kyle Cranmer, Eilam Gross, Ofer Vitells http://arxiv.org/abs/arXiv:1007.1727

This Demo has four parts:

1. TestJeffreysPriorDemo – validates Poisson mean case 1/sqrt(mu)
2. TestJeffreysGaussMean – validates Gaussian mean case
3. TestJeffreysGaussSigma – validates Gaussian sigma case 1/sigma
4. TestJeffreysGaussMeanAndSigma – demonstrates 2-d example

 
[#1] INFO:Minimization -- p.d.f. provides expected number of events, including extended term in likelihood.
[#1] INFO:Fitting -- RooAbsPdf::fitTo(p) fixing normalization set for coefficient determination to observables in data
[#1] INFO:Fitting -- using CPU computation library compiled with -mavx2
[#1] INFO:Fitting -- Creation of NLL object took 856.423 μs
[#1] INFO:Fitting -- RooAddition::defaultErrorLevel(nll_p_genData) Summation contains a RooNLLVar, using its error level
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: activating const optimization
Minuit2Minimizer: Minimize with max-calls 500 convergence for edm < 1 strategy 1
Minuit2Minimizer : Valid minimum - status = 0
FVAL  = -360.517018598809159
Edm   = 4.13877261906962124e-17
Nfcn  = 22
mu   = 100   +/-  9.98317  (limited)
[#1] INFO:Fitting -- RooAbsPdf::fitTo(p) Calculating sum-of-weights-squared correction matrix for covariance matrix
[#1] INFO:Minimization -- RooAbsMinimizerFcn::setOptimizeConst: deactivating const optimization
RooDataHist::genData[x] = 100 bins (100 weights)
 
  RooFitResult: minimized FCN value: -360.517, estimated distance to minimum: 3.66543e-18
                covariance matrix quality: Full, accurate covariance matrix
                Status : MINIMIZE=0 HESSE=0 HESSE=0 
 
    Floating Parameter    FinalValue +/-  Error   
  --------------------  --------------------------
                    mu    1.0000e+02 +/-  1.00e+01
 
variance = 100.016
stdev = 10.0008
jeffreys = 0.0999921
[#1] INFO:NumericIntegration -- RooRealIntegral::init(jeffreys_Int[mu]) using numeric integrator RooAdaptiveGaussKronrodIntegrator1D to calculate Int(mu)
[#1] INFO:NumericIntegration -- RooRealIntegral::init(Expected_Int[mu]) using numeric integrator RooIntegrator1D to calculate Int(mu)

 
#include "RooJeffreysPrior.h"
 
#include "RooWorkspace.h"
#include "RooDataHist.h"
#include "RooGenericPdf.h"
#include "RooPlot.h"
#include "RooFitResult.h"
#include "RooRealVar.h"
#include "RooAbsPdf.h"
#include "RooNumIntConfig.h"
 
#include "TH1F.h"
#include "TCanvas.h"
#include "TLegend.h"
#include "TMatrixDSym.h"
 
using namespace RooFit;
 
void rs302_JeffreysPriorDemo()
{
   RooWorkspace w("w");
   w.factory("Uniform::u(x[0,1])");
   w.factory("mu[100,1,200]");
   w.factory("ExtendPdf::p(u,mu)");
 
   std::unique_ptr<RooDataHist> asimov{w.pdf("p")->generateBinned(*w.var("x"), ExpectedData())};
 
   std::unique_ptr<RooFitResult> res{w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(true))};
 
   asimov->Print();
   res->Print();
   TMatrixDSym cov = res->covarianceMatrix();
   cout << "variance = " << (cov.Determinant()) << endl;
   cout << "stdev = " << sqrt(cov.Determinant()) << endl;
   cov.Invert();
   cout << "jeffreys = " << sqrt(cov.Determinant()) << endl;
 
   w.defineSet("poi", "mu");
   w.defineSet("obs", "x");
 
   RooJeffreysPrior pi("jeffreys", "jeffreys", *w.pdf("p"), *w.set("poi"), *w.set("obs"));
 
   RooGenericPdf *test = new RooGenericPdf("Expected", "Expected = 1/#sqrt{#mu}", "1./sqrt(mu)",
      *w.set("poi"));
 
   TCanvas *c1 = new TCanvas;
   RooPlot *plot = w.var("mu")->frame();
   pi.plotOn(plot);
   test->plotOn(plot, LineColor(kRed), LineStyle(kDashDotted));
   plot->Draw();
 
   auto legend = plot->BuildLegend();
   legend->DrawClone();
}
 
//_________________________________________________
void TestJeffreysGaussMean()
{
   RooWorkspace w("w");
   w.factory("Gaussian::g(x[0,-20,20],mu[0,-5.,5],sigma[1,0,10])");
   w.factory("n[10,.1,200]");
   w.factory("ExtendPdf::p(g,n)");
   w.var("sigma")->setConstant();
   w.var("n")->setConstant();
 
   std::unique_ptr<RooDataHist> asimov{w.pdf("p")->generateBinned(*w.var("x"), ExpectedData())};
 
   std::unique_ptr<RooFitResult> res{w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(true))};
 
   asimov->Print();
   res->Print();
   TMatrixDSym cov = res->covarianceMatrix();
   cout << "variance = " << (cov.Determinant()) << endl;
   cout << "stdev = " << sqrt(cov.Determinant()) << endl;
   cov.Invert();
   cout << "jeffreys = " << sqrt(cov.Determinant()) << endl;
 
   w.defineSet("poi", "mu");
   w.defineSet("obs", "x");
 
   RooJeffreysPrior pi("jeffreys", "jeffreys", *w.pdf("p"), *w.set("poi"), *w.set("obs"));
 
   const RooArgSet *temp = w.set("poi");
   pi.getParameters(*temp)->Print();
 
   //  return;
   RooGenericPdf *test = new RooGenericPdf("constant", "Expected = constant", "1", *w.set("poi"));
 
   TCanvas *c1 = new TCanvas;
   RooPlot *plot = w.var("mu")->frame();
   pi.plotOn(plot);
   test->plotOn(plot, LineColor(kRed), LineStyle(kDashDotted));
   plot->Draw();
 
   auto legend = plot->BuildLegend();
   legend->DrawClone();
}
 
//_________________________________________________
void TestJeffreysGaussSigma()
{
   // this one is VERY sensitive
   // if the Gaussian is narrow ~ range(x)/nbins(x) then the peak isn't resolved
   //   and you get really bizarre shapes
   // if the Gaussian is too wide range(x) ~ sigma then PDF gets renormalized
   //   and the PDF falls off too fast at high sigma
   RooWorkspace w("w");
   w.factory("Gaussian::g(x[0,-20,20],mu[0,-5,5],sigma[1,1,5])");
   w.factory("n[100,.1,2000]");
   w.factory("ExtendPdf::p(g,n)");
   //  w.var("sigma")->setConstant();
   w.var("mu")->setConstant();
   w.var("n")->setConstant();
   w.var("x")->setBins(301);
 
   std::unique_ptr<RooDataHist> asimov{w.pdf("p")->generateBinned(*w.var("x"), ExpectedData())};
 
   std::unique_ptr<RooFitResult> res{w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(true))};
 
   asimov->Print();
   res->Print();
   TMatrixDSym cov = res->covarianceMatrix();
   cout << "variance = " << (cov.Determinant()) << endl;
   cout << "stdev = " << sqrt(cov.Determinant()) << endl;
   cov.Invert();
   cout << "jeffreys = " << sqrt(cov.Determinant()) << endl;
 
   w.defineSet("poi", "sigma");
   w.defineSet("obs", "x");
 
   RooJeffreysPrior pi("jeffreys", "jeffreys", *w.pdf("p"), *w.set("poi"), *w.set("obs"));
 
   const RooArgSet *temp = w.set("poi");
   pi.getParameters(*temp)->Print();
 
   RooGenericPdf *test = new RooGenericPdf("test", "Expected = #sqrt{2}/#sigma", "sqrt(2.)/sigma", *w.set("poi"));
 
   TCanvas *c1 = new TCanvas;
   RooPlot *plot = w.var("sigma")->frame();
   pi.plotOn(plot);
   test->plotOn(plot, LineColor(kRed), LineStyle(kDashDotted));
   plot->Draw();
 
   auto legend = plot->BuildLegend();
   legend->DrawClone();
}
 
//_________________________________________________
void TestJeffreysGaussMeanAndSigma()
{
   // this one is VERY sensitive
   // if the Gaussian is narrow ~ range(x)/nbins(x) then the peak isn't resolved
   //   and you get really bizarre shapes
   // if the Gaussian is too wide range(x) ~ sigma then PDF gets renormalized
   //   and the PDF falls off too fast at high sigma
   RooWorkspace w("w");
   w.factory("Gaussian::g(x[0,-20,20],mu[0,-5,5],sigma[1,1.,5.])");
   w.factory("n[100,.1,2000]");
   w.factory("ExtendPdf::p(g,n)");
 
   w.var("n")->setConstant();
   w.var("x")->setBins(301);
 
   std::unique_ptr<RooDataHist> asimov{w.pdf("p")->generateBinned(*w.var("x"), ExpectedData())};
 
   std::unique_ptr<RooFitResult> res{w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(true))};
 
   asimov->Print();
   res->Print();
   TMatrixDSym cov = res->covarianceMatrix();
   cout << "variance = " << (cov.Determinant()) << endl;
   cout << "stdev = " << sqrt(cov.Determinant()) << endl;
   cov.Invert();
   cout << "jeffreys = " << sqrt(cov.Determinant()) << endl;
 
   w.defineSet("poi", "mu,sigma");
   w.defineSet("obs", "x");
 
   RooJeffreysPrior pi("jeffreys", "jeffreys", *w.pdf("p"), *w.set("poi"), *w.set("obs"));
 
   const RooArgSet *temp = w.set("poi");
   pi.getParameters(*temp)->Print();
   //  return;
 
   TCanvas *c1 = new TCanvas;
   TH1 *Jeff2d = pi.createHistogram("2dJeffreys", *w.var("mu"), Binning(10, -5., 5), YVar(*w.var("sigma"), Binning(10, 1., 5.)));
   Jeff2d->Draw("surf");
}

Author

Kyle Cranmer

Definition in file rs302_JeffreysPriorDemo.C.