Maximum Likelihood Method

Consider a set of variables $\vec{x}=(x_1, x_2)$. The decision to classify an event into one of the three classes $\ensuremath {{\cal{C}}}_1, \ensuremath {{\cal{C}}}_2, \ensuremath {{\cal{C}}}_3$ in based on the vector of measurements $\vec{x}$ and the joint PDFs $P_1(\vec{x}), P_2(\vec{x}), P_3(\vec{x})$. It is virtually impossible to decide the event class on an event-by-event basis. Hence we rely on a statistical analysis of the data to classify which events belong to $\ensuremath {{\cal{C}}}_1, \ensuremath {{\cal{C}}}_2,$ or $\ensuremath {{\cal {C}}}_3$ based on the measured $\vec{x}$ and our best approximation of the associated joint PDFs $P_1(\vec{x}), P_2(\vec{x}), P_3(\vec{x})$.

The random variables $x_i$ ($i$=1,2) for each class are distributed according to the joint PDF $P(\vec{x},\vec{\theta})$, with unknown parameters $\theta_1, \theta_2, \theta_3$ being the fraction of event of class $\ensuremath {{\cal{C}}}_1, \ensuremath {{\cal{C}}}_2, \ensuremath {{\cal{C}}}_3$. The extended maximum likelihood function [3] can be written as

$$\log L (\nu_{\rm {tot}}, \vec{\theta}) = -\nu_{\rm {tot}}(\v... ..._j + \theta_2 P_2 (\vec{x})_j + \theta_3 P_3 (\vec{x})_j] ,$$ (4)

where $\ensuremath {{\cal{N}}}$ is the number of bins, $n_j$ is the number of events in bin $j$, and the Poisson distributed variable for the total number of events is $\nu_{\rm {tot}}= n_{\rm {tot}} ( \theta_1 + \theta_2 + \theta_3 )$.