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Reference Guide
rs302_JeffreysPriorDemo.C File Reference

Detailed Description

View in nbviewer Open in SWAN 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
#include "RooWorkspace.h"
#include "RooDataHist.h"
#include "RooGenericPdf.h"
#include "RooPlot.h"
#include "RooFitResult.h"
#include "RooRealVar.h"
#include "RooAbsPdf.h"
#include "TH1F.h"
#include "TCanvas.h"
#include "TLegend.h"
#include "TMatrixDSym.h"
using namespace RooFit;
void JeffreysPriorDemo()
{
RooWorkspace w("w");
w.factory("Uniform::u(x[0,1])");
w.factory("mu[100,1,200]");
w.factory("ExtendPdf::p(u,mu)");
RooDataHist *asimov = w.pdf("p")->generateBinned(*w.var("x"), ExpectedData());
RooFitResult *res = w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(kTRUE));
asimov->Print();
res->Print();
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);
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();
RooDataHist *asimov = w.pdf("p")->generateBinned(*w.var("x"), ExpectedData());
RooFitResult *res = w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(kTRUE));
asimov->Print();
res->Print();
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);
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);
RooDataHist *asimov = w.pdf("p")->generateBinned(*w.var("x"), ExpectedData());
RooFitResult *res = w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(kTRUE));
asimov->Print();
res->Print();
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);
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);
RooDataHist *asimov = w.pdf("p")->generateBinned(*w.var("x"), ExpectedData());
RooFitResult *res = w.pdf("p")->fitTo(*asimov, Save(), SumW2Error(kTRUE));
asimov->Print();
res->Print();
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.