This tutorial explains the concept of global observables in RooFit, and showcases how their values can be stored either in the model or in the dataset.
Note: in this tutorial, we are multiplying the likelihood with an additional
likelihood to constrain the parameters with auxiliary measurements. This is
different from the rf604_constraints
tutorial, where the likelihood is
multiplied with a Bayesian prior to constrain the parameters.
With RooFit, you usually optimize some model parameters p
to maximize the
likelihood L
given the per-event or per-bin observations x
:
Often, the parameters are constrained with some prior likelihood C
, which
doesn't depend on the observables x
:
Usually, these constraint terms depend on some auxiliary measurements of
other observables g
. The constraint term is then the likelihood of the
so-called global observables:
For example, think of a model where the true luminosity lumi
is a
nuisance parameter that is constrained by an auxiliary measurement
lumi_obs
with uncertainty lumi_obs_sigma
:
As a Gaussian is symmetric under exchange of the observable and the mean parameter, you can also sometimes find this equivalent but less conventional formulation for Gaussian constraints:
If you wanted to constrain a parameter that represents event counts, you would use a Poissonian constraint, e.g.:
Unlike a Gaussian, a Poissonian is not symmetric under exchange of the observable and the parameter, so here you need to be more careful to follow the global observable prescription correctly.
Author: Jonas Rembser
This notebook tutorial was automatically generated with ROOTBOOK-izer from the macro found in the ROOT repository on Tuesday, March 19, 2024 at 07:17 PM.
import ROOT
Silence info output for this tutorial
ROOT.RooMsgService.instance().getStream(1).removeTopic(ROOT.RooFit.Minimization);
ROOT.RooMsgService.instance().getStream(1).removeTopic(ROOT.RooFit.Fitting);
l'(x | mu, sigma) = l(x | mu, sigma) * Gauss(mu_obs | mu, 0.2)
event observables
x = ROOT.RooRealVar("x", "x", -10, 10)
parameters
mu = ROOT.RooRealVar("mu", "mu", 0.0, -10, 10)
sigma = ROOT.RooRealVar("sigma", "sigma", 1.0, 0.1, 2.0)
Gaussian model for event observables
gauss = ROOT.RooGaussian("gauss", "gauss", x, mu, sigma)
global observables (which are not parameters so they are constant)
mu_obs = ROOT.RooRealVar("mu_obs", "mu_obs", 1.0, -10, 10)
mu_obs.setConstant()
note: alternatively, one can create a constant with default limits using RooRealVar("mu_obs", "mu_obs", 1.0)
constraint pdf
constraint = ROOT.RooGaussian("constraint", "constraint", mu_obs, mu, 0.1)
full pdf including constraint pdf
model = ROOT.RooProdPdf("model", "model", [gauss, constraint])
For most toy-based statistical procedures, it is necessary to also randomize the global observable when generating toy datasets.
To that end, let's generate a single event from the model and take the global observable value (the same is done in the RooStats:ToyMCSampler class):
dataGlob = model.generate({mu_obs}, 1)
Next, we temporarily set the value of mu_obs
to the randomized value for
generating our toy dataset:
mu_obs_orig_val = mu_obs.getVal()
ROOT.RooArgSet(mu_obs).assign(dataGlob.get(0))
Actually generate the toy dataset. We don't generate too many events, otherwise, the constraint will not have much weight in the fit and the result looks like it's unaffected by it.
data = model.generate({x}, 50)
When fitting the toy dataset, it is important to set the global observables in the fit to the values that were used to generate the toy dataset. To facilitate the bookkeeping of global observable values, you can attach a snapshot with the current global observable values to the dataset like this (new feature introduced in ROOT 6.26):
data.setGlobalObservables({mu_obs})
reset original mu_obs value
mu_obs.setVal(mu_obs_orig_val)
Create snapshot of original parameters to reset parameters after fitting
modelParameters = model.getParameters(data.get())
origParameters = modelParameters.snapshot()
When you fit a model that includes global observables, you need to specify them in the call to RooAbsPdf::fitTo with the RooFit::GlobalObservables command argument. By default, the global observable values attached to the dataset will be prioritized over the values in the model, so the following fit correctly uses the randomized global observable values from the toy dataset:
print("1. model.fitTo(*data, GlobalObservables(mu_obs))")
print("------------------------------------------------")
model.fitTo(data, GlobalObservables=mu_obs, PrintLevel=-1, Save=True).Print()
modelParameters.assign(origParameters)
1. model.fitTo(*data, GlobalObservables(mu_obs)) ------------------------------------------------ RooFitResult: minimized FCN value: 68.2482, estimated distance to minimum: 9.80327e-07 covariance matrix quality: Full, accurate covariance matrix Status : MINIMIZE=0 HESSE=0 Floating Parameter FinalValue +/- Error -------------------- -------------------------- mu 5.2717e-02 +/- 8.11e-02 sigma 9.7190e-01 +/- 9.73e-02
In our example, the set of global observables is attached to the toy dataset. In this case, you can actually drop the GlobalObservables() command argument, because the global observables are automatically figured out from the data set (this fit result should be identical to the previous one).
print("2. model.fitTo(*data)")
print("---------------------")
model.fitTo(data, PrintLevel=-1, Save=True).Print()
modelParameters.assign(origParameters)
2. model.fitTo(*data) --------------------- RooFitResult: minimized FCN value: 68.2482, estimated distance to minimum: 9.80327e-07 covariance matrix quality: Full, accurate covariance matrix Status : MINIMIZE=0 HESSE=0 Floating Parameter FinalValue +/- Error -------------------- -------------------------- mu 5.2717e-02 +/- 8.11e-02 sigma 9.7190e-01 +/- 9.73e-02
If you want to explicitly ignore the global observables in the dataset, you can do that by specifying GlobalObservablesSource("model"). Keep in mind that now it's also again your responsibility to define the set of global observables.
print('3. model.fitTo(*data, GlobalObservables(mu_obs), GlobalObservablesSource("model"))')
print("------------------------------------------------")
model.fitTo(data, GlobalObservables=mu_obs, GlobalObservablesSource="model", PrintLevel=-1, Save=True).Print()
modelParameters.assign(origParameters)
3. model.fitTo(*data, GlobalObservables(mu_obs), GlobalObservablesSource("model")) ------------------------------------------------ RooFitResult: minimized FCN value: 83.7181, estimated distance to minimum: 6.67911e-07 covariance matrix quality: Full, accurate covariance matrix Status : MINIMIZE=0 HESSE=0 Floating Parameter FinalValue +/- Error -------------------- -------------------------- mu 7.4744e-01 +/- 9.68e-02 sigma 1.2451e+00 +/- 1.38e-01
Draw all canvases
%jsroot on
from ROOT import gROOT
gROOT.GetListOfCanvases().Draw()