FIG. 6: (a) Prior densities computed using Method 1 (histogram) and Method 2 (dashed curve), and (b) corresponding posterior densities, for one of the channels in the D0 single top measurement and using 2.3 fb − 1 of data. For comparison we show the posterior density computed using a flat prior (dotted curve). of order 500 events, the precise form of the reference prior is not important. However, for small datasets — which is typical of searches for new or rare phenomena, one should expect the form of the prior to matter. It is then important to use a prior with provably useful properties, such as the ones enumerated in Sec. II. Our main purpose in this paper was to propose a set of formal priors with properties that make them attractive for use in the analysis of high energy physics data. Aside from the theoretical properties of invariance, coherence and sampling consistency, the reference prior method has three important practical advantages: (1) priors can be defined for almost any problem, regardless of the complexity of the likelihood function and the number of nuisance and interest parameters, (2) in contrast with flat priors, reference priors have so far always yielded proper posteriors, and (3) reference priors are computationally tractable, as shown by the single-top example. Here we have limited our numerical investigations to the class of likelihood functions that are derived from Poisson probability mass functions. For this class the Method-1 reference 24 VI. DISCUSSION  [pb] 0 2 4 6 8 ] -1 Prior density [pb 0.2 0.4 0.6 0.8 1 (a) Method 1 Method 2  [pb] 0 2 4 6 8 ] -1 Posterior density [pb 0.1 0.2 0.3 0.4 (b) Method 1 Method 2 Flat prior