Kyle Cranmer & Glen Cowan

License LGPL v2.1 (for ROOT compatibility, happy to make it BSD for other purposes)

This code quickly calculates expected discovery significance and upper limits based on an expected signal sExp, expected background, bExp, and uncertainty on the background estimate, deltaB. In this situation, most of the problem can be done analytically, and no minimization algorithms are necessary. Observed versions and +/- 1,2 sigma bands on the expected limit are in the works.

The expected 95% CLs upper limit on the number of signal events, s, given an expected background, bExp, and uncertainty on the background estimate, deltaB. The background uncertainty is absolute (not relative) and is uncertainty on the mean background (so you don't include sqrt(bExp) Poisson fluctuations in this number).

Example Usage:

• you expect 50 +/- 7 background events
• ExpectedLimit(50,7)
• returns s_95 = 19.7 events

Similar code for expected discovery significance is also included.

Example Usage:

• you expect 50 signal events, 100 +/- 7 background events
• ExpectedSignificance(50,100,7)
• returns 3.72 sigma

The derivations of these formulae are based on a statistical model:

Pois(n | s+b ) * Pois(m | tau * b)

The first term is the standard "number counting" signal region. The second term is a idealized auxiliary measurement used to constrain the background resulting in some characteristic relative uncertainty. The tau quantity is calculated from tau=bExp/deltaB/deltaB. See arXiv:physics/0702156 for motivation of this model. The maximum likelihood estimate and conditional maximum likelihood estimate were solved analytically and coded here. The log-likelihood ratio and the profile log likelihood ratio follow immediately. The b-only and s+b p-values needed for CLs are calculated using the asymptotic distributions in

Cowan, Cranmer, Gross, Vitells, Eur. Phys. J. C 71 (2011) 1554.
arXiv:1007.1727

The ExpectedSignificance formulae was derived by Cowan and the numerical solution for the Expected upper limit was written by Cranmer as correlaries to that paper.

Note, the ExpectedSignificance equation is the same as Eq.(17) of

Tipei Li and Yuqian Ma, Astrophysical Journal 272 (1983) 317–324.

and Eq.(25) of

Robert D. Cousins, James T. Linnemann and Jordan Tucker, NIM A 595 (2008) 480– 501; arXiv:physics/0702156.

after making the replacements n=bExp and m=bExp*tau.

This code could use more validation, but there are a series of tests that run that check against results from the mathematica notebook and the equivalent HistFactory example. The HistFactory validation is in the directory validation and includes:

• A HistFactory model in xml that defines the statistical model above
• The top-level script makeHists.C that writes histograms to file data/histograms.root based on the values of bExp and deltaB, then runs HistFactory's hist2workspace + RooStats/StandardHypoTestInvDemo.C with the AsymptoticCalculator using the 1-sided profile likelihood ratio test statistic and CLs=True.

This is a work in progress!

• C++ to be updated to use Brent q root finding instead of simple scan
• remove ROOT dependency in C++ verison entirely?
• add +/- 1,2 sigma bands for the expected upper limit
• could do same for observed upper limit.
• add a Mathematica version

Extension:

• Include uncertainty on signal efficiency for upper limit. Those equations are big! (See ExpectedLimitWithSignalUncert.nb)