content_image/352139.jpg content_image/352140.jpg
⃗ n generated from p ( ⃗ n | σ ), of an accurate numerical approximation of the second derivative of the negative log-likelihood [20]. As already pointed out, Method 2 does not require a compact subset normalization procedure. The reference posterior is thus proportional to the product of p ( ⃗ n$_{o}$ | σ ) and π$_{R}$$_{2}$ ( σ ), and the normalization with respect to σ must be determined numerically. 

## IV. VALIDATION STUDIES 

We have performed a number of studies to validate inferences from the single-count model, using both the numerical algorithms described in Sec. III C and analytical expressions we obtained for the marginal Method-1 and 2 posteriors for σ . To recapitulate, we have two reference priors for this model: 

<img src='content_image/352139.jpg'>

<img src='content_image/352140.jpg'>

and we have assumed that π ( ϵ, µ | σ ) = π ( ϵ, µ ) at Eq. (17). As explained in Sec. III, this extra assumption affects only the definition of π$_{R}$$_{2}$ , which therefore incorporates more information than π$_{R}$$_{1}$ . In the present section we study and compare the properties of these two reference priors. To begin, we show some example prior and posterior σ marginals in Fig. 2. As expected, posteriors corresponding to a small observed number of events favor small cross sections, and posteriors derived from flat priors put less weight on small cross sections than reference posteriors. 

Our derivations of the two reference prior methods made use of Jeffreys’ rule (15). As pointed out in Sec. II B, this approach assumes that some regularity conditions are satisfied, such that the resulting posterior is asymptotically normal. We now wish to verify this assumption with a graphical example. If one adopts the objective Bayesian view that the parameters σ , ϵ , and µ have true values, then the asymptotic limit can be defined as the result of a large number N$_{R}$ of replications of the measurement, in the limit where that number goes to infinity. For the case where each measurement replication i consists of a number of events n$_{i}$ drawn from a probability mass function f ( n | σ, ϵ, µ ), the reference 

