Upper Limits on Poisson Processes

This note presents a technique for the calculation of upper limits on Poisson processes at some desired conndence level given a certain number of observed events, an expected number of background events, and uncertainty on both the overall acceptance and the expected background. This hybrid frequentist/Bayes technique represents the standard adopted by the CDF Collaboration in presenting the results of new particle search counting experiments. A \fully Bayesian" approach incoporating uncertainties yields limits which match the frequentist limits in the zero-uncertainty case, but otherwise exceeds the frequentist limits. 1 Motivation In many, if not most, new particle searches in high energy physics, one selects from a large number of recorded events those which bear characteristics of the new process while minimizing the retention of events from well-understood processes. This typically results in a small number of events passing the selection requirements, consistent with the expectation from a calculation of the expected background. At this stage one can determine an upper limit on the number of signal events present in the sample, at some desired conndence level (usually 95%), employing the method described by the Particle Data Group (PDG). 1]. This \PDG" method, however, is silent on the issue of how to incorporate the uncertainties present in the expected number of background events and in the overall acceptance. Other authors 2, 3] have separately discussed methods for incorporating uncertainties in the background or acceptance, but none to date (to our knowledge) have presented a method incorporating both. This note discusses the method used by the CDF Collaboration to determine upper limits on Poisson processes in the presence of uncertainties (both statistical and systematic) simultaneously in the accpetance and background, essentially merging the approaches of Zech and 1 Huber et al. referred to above. Though often thought of as an example of a classical (\frequen-tist") statistical procedure, this method in fact uses Bayesian notions of a probability density function (pdf) for the unknown true values of the acceptance and expected background. In the case of no uncertainties in the acceptance or expected background, the full-blown Bayesian treatment gives the same upper limits as the frequentist PDG method..3] As shown below, though, when uncertainties are incorporated into the fully Bayesian algorithm, the resulting limits exceed those obtained in the frequentist method. This note should serve as a reference for this method for those who wish to cross check their own calculations or wish …