If the total statistical uncertainty in bin b is defined as delta_b, and the MC yield in that bin is nu_b, the relative uncertainty should be (delta_b/nu_b)? (The text has the reciprocal of this, as a typo I suspect).
Then the equivalent yield which has the same relative stat. unc. as above should be m_b = (nu_b/delta_b)^2 (Again, the text has the reciprocal).
Finally, the text defines another parameter called tau_b = (nu_b/delta_b)^2. To me this is the same as what m_b is supposed to be, and I don’t see the point of this new parameter. If so, the Poisson constraint term (the second term in eqn 9) becomes
Pois(m_b | gamma_b * m_b)
meaning that the “expected” value of gamma_b is 1. Am I understanding this correctly?
I am writing not to criticize the helpful paper, but rather to make sure I understand the math.
I tried to read through and understand the Beeston Barlow code (ROOT: roofit/histfactory/src/RooBarlowBeestonLL.cxx Source File), and I see that the code does have two separate parameters for m_b and tau_b, and gets them in different ways. But it is not immediately clear to me what the difference between these two parameters is.
I think you are right, in the text there is a wrong definition of m_b. Actually the definition of tau_b is correct, m_b is a global observable, corresponding to the amount of total MC bin sample size.
The gamma_b is just a factor, centred around 1 to express the uncertainty of this MC bin size. It will come in the likelihood with a corresponding constraint term. The value of tau_b is instead the expected contribution of the MC and it is considered fixed.
I hope this clarify the difference between m_b , gamma_b and tau_b