content_image/370559.jpg
of π ( φ | θ ), then it will seem preferable to use Method 1, which only requires knowledge of the marginal prior π ( φ ). When one is unsure of which method to use, one should use both, and treat the results as part of a test of robustness. An important practical advantage of Method 1 is that the conditional reference prior is computed once and for all, for a given model, and can be used with any evidence-based prior for the nuisance parameters. In contrast, for Method 2 the reference prior must be computed anew every time the priors for the nuisance parameters change. On the other hand, since Method 2 reduces the problem to one involving a single parameter, the reference prior algorithm reduces to Jeffreys’ rule (15), which is typically easier to implement. 

In the next section we introduce the basic model studied in this paper, and follow with the application of Methods 1 and 2 to that model. 

## A. The Single-Count Model 

A very common model for high energy physics measurements is the following. A number of events N is observed by some apparatus, and it is assumed that N is Poisson distributed with mean count ϵ σ + µ , where σ is the rate of a physics signal process, typically the cross section, which we detect with an effective integrated luminosity ϵ — that is, the integrated luminosity scaled by the signal efficiency, and µ is a background contamination. Thus, σ is the parameter of interest, whereas ϵ and µ are nuisance parameters for which we usually have partial information. For physical reasons none of these three parameters can be negative. We write the likelihood for this model as 

<img src='content_image/370559.jpg'>

Information about ϵ and µ usually comes from a variety of sources, such as auxiliary mea- surements, Monte Carlo simulations, theoretical calculations, and evidence-based beliefs (for example, some sources of background contributing to µ may be deemed small enough to ig- nore, and some physics effects on ϵ , such as gluon radiation, may be believed to be well enough reproduced by the simulation to be reliable “within a factor of 2”). It is therefore natural to represent that information by an evidence-based prior. Here we will assume that ϵ and µ are independent of σ and that their prior factorizes as a product of two gamma 

