1. CMS RooStats Higgs Combination Package
Giovanni Petrucciani (UCSD)

2. the combination package
A package that builds an executable that can be run to compute limits/signif.
combine input –M method [options]
Input can be a text “datacard” or an arbitrary RooStats Workspace saved in a ROOT file.
Internally, it has two main components
text2workspace: A python program that reads datacards and creates a RooStats model
The part that deals with the statistical methods

3. Datacards
Started from a simple format for counting experiments: a text table with observed events and expected yields in each channel.
Progressively enhanced it:
Support for more pdfs for systematic uncertainties (e.g. Gammas, asymm errors, ...)
Use of shapes instead of just counting:
Binned shapes: plain ROOT histograms
Arbitrary shapes (RooAbsPdf)
RooDataHist and RooDataSet (or just TTrees)

4. Simple datacards example
# one channel, we observe 0 events bin 1 observation # expected events for signal and backgrounds bin 1 1 process ggh4G Bckg process 0 1 rate deltaS lnN % unc. on signal deltaB lnN % unc. on bkg.

5. Complex counting experiment
bin e_tau mu_tau e_mu observation bin e_tau e_tau e_tau mu_tau mu_tau mu_tau e_mu e_mu e_mu process higgs ZTT QCD higgs ZTT QCD higgs ZTT other process rate lumi lnN tauid lnN ZtoLL lnN effic lnN QCDel lnN QCDmu lnN other lnN
Multiple channels
Treating of individ. backgrounds
Correlated effects of systematics
Systematics with names, to allow combination of datacards

6. From Counting to Shapes
For number countings, the Likelihood in each channel is constructed as sum over all the contributing physics processes
N(exp) = sum( N(exp, proc), … )
pdf = Poisson(N(obs) | N(exp))
full pdf = PROD(pdf(channel1), pdf(channel2), …)
The extension to shapes is trivial
pdf = RooAddPdf( N(exp, proc) * shape(proc), … )
full pdf = RooSimultaneous( pdf1, pdf2, … )

7. Shape Uncertainties (1)
Vertical Morphing:
Using linear or quadratic interpolation between 3 templates.
Gaussian contraint on interpolation parameter.
Works on any pdf: histo, keys, parametric
Gives wrong results if the deformation is large (e.g. for a gaussian if the uncertainty on the mean is comparable with the peak width)

8. Shape Uncertainties (2)
Simple parametric uncertainties:
Shape parameter used as nuisance, with a constraint (Gaussian or Bifurcated Gaussian
Relies on the user to select a physics-oriented parametrization in which nuisances are uncorrelated.

9. Shape uncertainties (3)
Parametric interpolation:
Templates are created, e.g. from different MCs
Same parametrization is used to fit all templates, and save two additional RooArgSets of parameters for each systematic
Shapes are obtained interpolating linearly the parameters between the RooArgSets
(still under developed, not yet there)

10. Combined Datasets
If all channels are counting experiments, make a RooDataSet with one column per channel
If all channels have all inputs as TH1s (data and templates), convert into RooDataHist all on the same dummy variable. Histos for channels that have less bins get padded with empty bins.
If at least one channel has a RooDataSet dataset, try to make a combined RooDataSet
Otherwise try to make a combined RooDataHist

11. Statistical tools proper
What we do on top of RooStats:
Configure RooStats components through command-line args and sensible defaults
Extend functionalities of some tools, or provide workarounds to bugs for which the fix is not in yet in CMSSW’s build of RooStats
Provide other common infrastructure, e.g.
Running toys to get expected limits
Saving results to root files

12. Zoo of statistical methods
Asymptotic limits and significances from profiled likelihood
Bayesian limits
Various flavours of Hybrid Frequentist-Bayesian limits (“CLs”, ...) and significances
Feldman-Cousins bands
Computing both observed limits and expected limits with 68%/95% bands by running toys.

13. Other tools Other pieces: Tools to combine datacards
Tools to run the combination across the GRID using CRAB (the CMS grid wrapper).
Plotting tools: basic functionality exists, but something better is under work

14. What’s higgs specific?
Datacard format: generic “excess of signal on top of some background” problem.
Statistical tools even more generic than this.
Main limitations:
Most things work only when setting a limit in a single parameter of interest
Parameter should behave like a cross section (positive definite; zero = no signal)
Limitations driven by lack of need for something even more general; can be overcome if needed.

15. CMS dependencies text2workspace: combination tool:
AsymPow for asymmetric log-normals
VerticalInterpPdf for vertical morphing
combination tool:
hypo test inversion with new HybridCalculator
optimized TestStatistics classes
some helper functions e.g. to factorize pdfs

16. External dependencies
text2workspace:
python libraries (re, optparse) + PyROOT
combination tool:
c++ std library (io, stl, auto_ptr, exceptions)
boost::program_options used to parse command-line args and configure methods
boost::filesystem to do a “rm –r” of demporary dir
some system-level stuff (e.g. fork, tempnam)