Likelihood ratio intervals with Bayesian treatment of uncertainties: coverage, power and combined experiments

In this note we present studies of coverage and power for confidence intervals for a Poisson process with known background calculated using the Likelihood ratio (aka Feldman & Cousins) ordering with Bayesian treatment of uncertainties in nuisance parameters. We consider both the variant where the Bayesian integration is done in both the numerator and the denominator and the modification where the integration is done only in the numerator whereas in the denominator the likelihood is taken at the maximum likelihood estimate of the parameters. Furthermore we discuss how measurements can be combined in this framework and give an illustration with limits on the branching ratio of a rare B-meson decay recently presented by CDF/D0. A set of C++ classes has been developed which can be used to calculate confidence intervals for single or combining multiple experiments using the above algorithms and considering a variety of parameterizations to describe the uncertainties.