improper. However, for small acceptance uncertainties the divergence of the upper limit is often concealed by the inevitable truncation of numerical computations [6]. 

The specification of priors that encode minimal information is of such importance in practice that a large body of literature exists describing attempts to construct priors that yield results with provably useful characteristics. These priors are typically arrived at using formal rules. In this paper, therefore, we refer to them as formal priors [7] to distinguish them from evidence-based priors. Many such formal rules exist [4]. In this paper we study, and then recommend, a rule which is arguably the most successful: that developed by Bernardo [8] and Berger and Bernardo [9–11]. Formal priors constructed according to the Bernardo-Berger rule are called reference priors , a somewhat unfortunate name given that the term reference prior is sometimes used as a synonym for what we have called a formal prior. 

Reference priors have been shown to yield results with several desirable properties, all of which should appeal to particle physicists. Therefore, in principle such priors could be a foundation for Bayesian inference in particle physics research. However, reference priors and the associated methods collectively referred to as reference analysis [12, 13] have yet to enter the field in a significant way. The purpose of this paper is to initiate this process by applying the Bernardo-Berger method to a familiar, but important class of problems, namely that of calculating posterior densities for signal cross sections. 

In the next section we describe the general goals of reference prior construction and show how these are implemented via the concept of missing information. For simplicity we limit that discussion to one-parameter problems. Section III then considers the treatment of nui- sance parameters about which prior information is available. Examples of such parameters include detector calibration constants, background contaminations, geometrical acceptances, and integrated luminosities. We describe two methods for handling these parameters, de- pending on the type of information that is available about them. These methods are then applied to counting experiments with uncertain background contamination and effective lu- minosity. In the simplest cases we have obtained analytical expressions for the marginal posterior for the quantity of interest. For the general case we have developed a numerical algorithm. Some appealing properties of these posteriors are examined in Sec. IV. In Sec. V the reference prior methodology is applied to a recent measurement of the production cross section for single top quarks at the Tevatron. Final comments are presented in Sec. VI. 

