Hi ROOT experts,
I have been studying the implementation of the weighted log-likelihood fit (“WL” option) in ROOT (v6.36.04), and I would like to understand the statistical motivation behind one part of the implementation.
From the source code, ROOT defines the effective scaling factor as
s_i = sigma_i^2 / y_i
and
n_eff = y_i / s_i = y_i^2 / sigma_i^2
mu_eff = mu_i / s_i
which gives
l_i = s_i * [ n_eff * log(n_eff / mu_eff) + mu_eff - n_eff ]
I have independently implemented this expression and verified numerically that it reproduces ROOT’s MinFCN exactly for my weighted histogram fits.
While studying this implementation, I became curious about the role of the final scaling factor.
Once the effective quantities are defined as
n_eff = y_i / s_i
mu_eff = mu_i / s_i
it seems natural to consider the effective Poisson deviance
l_i = n_eff * log(n_eff / mu_eff) + mu_eff - n_eff by itself.
However, ROOT instead minimizes
l_i = s_i * [ n_eff * log(n_eff / mu_eff) + mu_eff - n_eff ]
While studying this implementation, I became curious about the role of the final scaling factor.
The effective count formalism is also discussed in the paper G. Böhm and G. Zech, Statistics of weighted Poisson events and its applications.
In a section on the Scaled Poisson Distribution (SPD), it is introduced for parameter estimation, where the effective quantities are defined in essentially the same way as in ROOT.
However, the likelihood in the paper does not appear to include the additional scaling factor s_i that multiplies the effective Poisson deviance.
So my question is:
Why does ROOT include the final scaling factor s_i?
-
If the effective quantities (
n_effandmu_eff) are already introduced, why is the effective Poisson deviance by itself not sufficient? -
Would omitting the factor
s_ilead to an incorrect likelihood or estimator? If so, could you explain what statistical property would be lost?
If there is a derivation, technical note, or publication explaining the origin of this factor, I would greatly appreciate a reference.
