content_image/373021.jpg content_image/373020.jpg content_image/373019.jpg
## B. The Multiple-Count Model 

An important generalization of the single-count model is obtained by considering M replications of the latter; the likelihood is: 

<img src='content_image/373021.jpg'>

To obtain the Method 1 reference prior for this model, we first calculate Jeffreys’ prior for σ , while keeping ⃗ ϵ and ⃗ µ fixed: 

<img src='content_image/373020.jpg'>

This prior is improper, requiring us to apply the compact subset normalization described in Sec. III A 1. Using a straightforward generalization of the nested compact sets of Eq. (24), we find that the correct reference prior is identical to Jeffreys’ prior. 

In order to apply Method 2, we need to specify a proper conditional prior for the µ$_{i}$ and ϵ$_{i}$ given σ . Neglecting correlations, we set: 

<img src='content_image/373019.jpg'>

The marginalized data probability distribution p ( ⃗ n | σ ) is then a product of expressions of the form (28), one for each count i . 

Here we no longer attempt to obtain analytical expressions for the Method 1 and 2 reference posteriors. Instead, we use the numerical algorithms described below. 

## C. Numerical Algorithms 

In this section we describe numerical algorithms that can be used to compute Method 1 or 2 reference posteriors for the single- and multiple-count Poisson likelihoods discussed in the previous sections. 

For Method 1 the algorithm starts by generating ( σ, ⃗ ϵ, ⃗ µ ) triplets from the “flat-prior posterior”, i.e. the posterior obtained by setting π ( σ | ⃗ ϵ, ⃗ µ ) = 1 (line 3 in the pseudo-code below); the correct reference prior π ( σ | ⃗ ϵ, ⃗ µ ) is then computed at lines 4–7 and is used at line 9 to weight the generated σ values so as to produce the reference posterior: 

