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: To obtain the Method 1 reference prior for this model, we first calculate Jeffreys’ prior for σ , while keeping ⃗ ϵ and ⃗ µ fixed: 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: 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. 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: C. Numerical Algorithms 16 π ( ⃗ ϵ, ⃗ µ | σ ) = M ∏ i =1 a$_{i}$ ( a$_{i}$ϵ$_{i}$ ) x i − 1 / 2 e − a i ϵ i Γ( x$_{i}$ + 1 / 2) b$_{i}$ ( b$_{i}$µ$_{i}$ ) y i − 1 / 2 e − b i µ i Γ( y$_{i}$ + 1 / 2) . (34) π$_{J}$ ( σ | ⃗ ϵ, ⃗ µ ) ∝ √ M ∑ i =1 ϵ 2 i ϵ$_{i}$ σ + µ$_{i}$ . (33) p ( ⃗ n | σ, ⃗ ϵ, ⃗ µ ) = M ∏ i =1 ( ϵ$_{i}$σ + µ$_{i}$ ) n i n$_{i}$ ! e − ϵ i σ − µ i . (32)