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:
Author: Kyle Cranmer
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Thursday, June 30, 2022 at 09:51 AM.
%%cpp -d
#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"
%%cpp -d
// This is a workaround to make sure the namespace is used inside functions
using namespace RooFit;
%%cpp -d
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();
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();
}
%%cpp -d
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();
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();
}
%%cpp -d
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();
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");
}
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();
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();
Draw all canvases
%jsroot on
gROOT->GetListOfCanvases()->Draw()