Furthermore, this specification must be done conditionally on the signal rate σ . As men- tioned earlier, we will use expression (17) for this prior. The next step in the application of Method 2 is to marginalize the probability model (16) with respect to ϵ and µ : where and the binomial coefficients are expressed in terms of gamma functions to accomodate noninteger values of their arguments. Finally, the reference prior algorithm must be applied to the marginalized model p ( n | σ ). As in the case of Method 1, the conditions for applying Jeffreys’ rule are satisfied here; we therefore obtain: We will use the notation π$_{R}$$_{2}$ ( σ ) to refer to the marginal reference prior for σ obtained with Method 2. Note that the compact subset argument invoked in the construction of the Method 1 reference prior is not needed here because all the parameters other than σ have already been eliminated by marginalization. For Method 2 the marginal reference posterior for σ is proportional to the product of the marginal data probability distribution (28) and the marginal reference prior (30): The normalization of π$_{R}$$_{2}$ ( σ | n ) must be obtained numerically. 15 p ( n | σ ) = ∫∫ p ( n | σ, ϵ, µ ) π ( ϵ, µ | σ ) dϵ dµ, = ∫∫ ( ϵσ + µ ) n n ! e − ϵσ − µ a ( aϵ ) x − 1 2 Γ( x + 1 $_{2}$) e − aϵ b ( bµ ) y − 1 2 Γ( y + 1 $_{2}$) e − bµ dϵ dµ, = [ a a + σ ] x + 1 2 [ b b + 1 ] y + 1 2 S 0 $_{n}$( σ ) , (28) S m n ( σ ) ≡ n ∑ k =0 k m ( k + x − 1 2 k ) ( n − k + y − 1 2 n − k ) [ 1 b + 1 ] n − k [ σ a + σ ] k , (29) π$_{R}$$_{2}$ ( σ ) ∝ √ ∞ ∑ n =0 [ ( x + 1 $_{2}$) S 0 $_{n}$( σ ) − a σ S 1 $_{n}$( σ ) ] 2 ( a + σ ) x +5 / 2 S 0 $_{n}$( σ ) . (30) π$_{R}$$_{2}$ ( σ | n ) ∝ p ( n | σ ) π$_{R}$$_{2}$ ( σ ) . (31)