densities: where a , b , x , and y are known constants, related to the means ¯ ϵ , ¯ µ and coefficients of variation δϵ , δµ by: The built-in assumption that ϵ and µ are uncorrelated is clearly an approximation, since they share a dependence on the integrated luminosity, which is itself uncertain. There are two ways of interpreting this prior. The first one is appropriate when informa- tion about ϵ and µ comes from one or more non-experimental sources, such as Monte Carlo studies and theoretical calculations, and takes the form of a central value plus an uncertainty. Since the ϵ and µ components of the prior are each modeled by a two-parameter density, one can fix the shape of this density in each case by matching its mean with the central value of the corresponding measurement and its standard deviation with the uncertainty. It will then be necessary to check the robustness of the final analysis results to reasonable changes in this procedure. For example, one may want to replace the gamma distribution by a log-normal or truncated Gaussian one, and the mean by the mode or median. The second interpretation of prior (17) follows from the analysis of two independent, aux- iliary Poisson measurements, in which the observed number of events is x for the effective luminosity and y for the background. The expected numbers of events in these auxiliary measurements are aϵ and bµ , respectively. For a Poisson likelihood with mean aϵ the refer- ence prior coincides with Jeffreys’ prior and is proportional to 1 $_{/}$√$_{ϵ}$$_{. Given a measurement}$ x , the posterior will then be a gamma distribution with shape parameter x + 1 / 2 and scale parameter 1 /a . A similar result holds for the background measurement. In this manner the prior (17) is obtained as a joint reference posterior from two auxiliary measurements. The problem we are interested in is finding a prior for σ , about which either little is known or one wishes to act as if this is so. This section serves two purposes: to illustrate the analytical algorithm for computing reference priors and to apply Method 1 to model (16). 12 1. Application of Method 1 to the Single-Count Model π ( ϵ, µ | σ ) = π ( ϵ, µ ) = a ( aϵ ) x − 1 / 2 e − aϵ Γ( x + 1 / 2) b ( bµ ) y − 1 / 2 e − bµ Γ( y + 1 / 2) , (17) ¯ ϵ = x + 1 2 a , δϵ = 1 √ x + 1 2 , ¯ µ = y + 1 2 b , δµ = 1 √ y + 1 2 . (18)