Detailed Description

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.

Introduction

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:

\[ L( x | p ) \]

Often, the parameters are constrained with some prior likelihood C, which doesn't depend on the observables x:

\[ L'( x | p ) = L( x | p ) * C( p ) \]

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:

\[ L'( x | p ) = L( x | p ) * C( g | p ) \]

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:

\[ L'(data | mu, lumi) = L(data | mu, lumi) * \text{Gauss}(lumi_obs | lumi, lumi_obs_sigma) \]

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:

\[ L'(data | mu, lumi) = L(data | mu, lumi) * \text{Gauss}(lumi | lumi_obs, lumi_obs_sigma) \]

If you wanted to constrain a parameter that represents event counts, you would use a Poissonian constraint, e.g.:

\[ L'(data | mu, count) = L(data | mu, count) * \text{Poisson}(count_obs | count) \]

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.

 
 
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);
 
# Setting up the model and creating toy dataset
# ---------------------------------------------
 
# 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])
 
# Generating toy data with randomized global observables
# ------------------------------------------------------
 
# 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)
 
# Fitting a model with global observables
# ---------------------------------------
 
# 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)
 
# 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)
 
# 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)

 
  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
 
 
  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
 
 
  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
 
1. model.fitTo(*data, GlobalObservables(mu_obs))
------------------------------------------------
2. model.fitTo(*data)
---------------------
3. model.fitTo(*data, GlobalObservables(mu_obs), GlobalObservablesSource("model"))
------------------------------------------------

Date: January 2022

Author: Jonas Rembser

Definition in file rf613_global_observables.py.