content_image/335278.jpg content_image/335279.jpg content_image/335280.jpg content_image/335281.jpg
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 : 

<img src='content_image/335278.jpg'>

where 

<img src='content_image/335279.jpg'>

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: 

<img src='content_image/335280.jpg'>

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. 

<img src='content_image/335281.jpg'>

