quentist ones, and a proper evaluation requires first of all the specification of an ensemble of experiments. A well-known property of Bayesian posterior intervals constructed from a proper prior is that their coverage is exact when averaged over the prior [21]. This is an immediate consequence of the law of total probability. Indeed, given a parameter θ with proper prior π ( θ ), and a measurement X , the prior-averaged frequentist coverage of a 1 − α Bayesian credibility interval R ( X ) can be written as: where the first expectation is over the prior π ( θ ) and the second one over the marginal sampling distribution m ( x ) = ∫ p ( x | θ ) π ( θ ) dθ . When π ( θ ) is a reference prior, and especially when it is improper, there is no natural metric over which the coverage can be averaged. The only sensible approach in that case is to study the coverage pointwise, i.e. as a function of the true value of θ . Since the single- and multiple-count models discussed in this paper combine an improper prior for the parameter of interest σ with proper priors for the nuisance parameters ⃗ ϵ and ⃗ µ , we will study interval coverage for a fixed value of σ , but averaged over π ( ⃗ ϵ, ⃗ µ ). Our interest is in how this coverage evolves toward the asymptotic limit. As before, we take this limit in the sense of an ever-increasing number N$_{R}$ of experiment replications. For a given value of N$_{R}$ , the posterior is formed as in Eq. (37) and its coverage is computed. Figure 4 shows the coverage of 95% credibility upper limits and 68% credibility central intervals as a function of N$_{R}$ . As the latter increases, the coverage converges to the credibility, confirming the sampling consistency of the method. Finally, we examine the behavior of reference posterior upper limits on σ as a function of the expected background ¯ µ (defined in Eq. (18)) when the observed number of events n is small. For comparison, when n = 0 and there are no uncertainties on signal efficiency and background, frequentist upper limits decrease linearly with background. From a Bayesian point of view this result is surprising. Indeed, when zero events are observed the likeli- hood function factorizes exactly into background and signal components, indicating that the experiment actually performed can be analyzed as the combination of two independent experiments, one to measure background and the other signal. If, in addition, signal and background are a priori independent, then posterior inferences about signal will be inde- pendent of background. In particular, upper limits on σ will be constant as a function of ¯ µ , not linearly decreasing. The reference priors entangle signal and background however, so 21 E$_{π}$ [ P ( θ ∈ R ( X ) | θ )] = P ( θ ∈ R ( X )) = E$_{m}$ [ P ( θ ∈ R ( X ) | X )] = E$_{m}$ (1 − α ) = 1 − α, (38)