# Extended likelihood method
- In particle physics, we care about the signal, background yields $\hat{S}, \hat{B}$ in one measurement.
- The signal, background events follow different PDFs denoting as $\mathcal{P}_{S}(X_{i}; \theta), \mathcal{P}_{B}(X_{i}; \theta)$:
> where $X_{i}$ is the random variable we collect in an experiment, $\theta$ can be some known truth parameter or unkown parameter to be measured.

\begin{equation}
\begin{aligned}
\mathcal{L}(X_{i}; \hat{S}, \hat{B}, \theta)
&= \frac{(\hat{S}+ \hat{B})^Ne^{-(\hat{S}+\hat{B})}}{N!} \prod_{i=0}^{N} \{ \frac{\hat{S}}{N} \mathcal{P}_{S}(X_{i}; \theta) + \frac{\hat{B}}{N} \mathcal{P}_{B} (X_{i}; \theta)\} \\
&= \frac{e^{-(\hat{S}+\hat{B})}}{N!} \prod_{i=0}^{N} \{ \hat{S} \mathcal{P}_{S}(X_{i}; \theta) + \hat{B} \mathcal{P}_{B} (X_{i}; \theta)\}
\end{aligned}
\tag{1}
\end{equation}

- Question:
This likelihood is not like the classic MLE exercise, which is by multiplying the probabilities of all random variables (w/ presumed PDFs). A poisson probability is introduced to estimate how likely we get certain $\hat{S}, \hat{B}$ in the measurement. This is a technique used frequently in our field.
Is it valid to say that for each random variable, we are not sure which PDF it follows, so we do a conditional probability calculation: $\frac{\hat{S}}{N} \mathcal{P}_{S}(X_{i}; \theta) + \frac{\hat{B}}{N} \mathcal{P}_{B} (X_{i}; \theta)$?
> Then why we need the poisson term (fraction)

- Analogy:
假設有 A, B 兩班學生，身高(RV)分佈不同，今天兩班一起測量身高，想要去估計 A, B 班各有多少學生
# Combined datasets to measure some parameters
In practice, we want to measure the **truth happen rate** (we call it branching fraction **$BF$**) of signal event, and we relate the $BF$ and $S$ by given equation ($BF = \frac{S}{\epsilon \times N(B\bar{B})}$), so we'd would like to rewrite the Eq(1) into:
\begin{equation}
\begin{aligned}
\mathcal{L}(X_{i}; \hat{BF}, \hat{B}, \theta)
\end{aligned}
\tag{2}
\end{equation}
And we consider independent datasets, with different PDFs {$\mathcal{P}_{1,S}, \mathcal{P}_{1,B}$}, {$\mathcal{P}_{2,S}, \mathcal{P}_{2,B}$}. The two datasets share a same parameter $BF$.
\begin{equation}
\begin{aligned}
BF = \frac{S_{1}}{\epsilon_{1} \times N_{1}(B\bar{B})} = \frac{S_{2}}{\epsilon_{2} \times N_{2}(B\bar{B})}
\end{aligned}
\end{equation}
Can we construct a joint likelihood $\mathcal{L}_{tot}(X_{i}) = \mathcal{L}_{1}(X_{i})\mathcal{L}_{2}(X_{i})$ (is this formulation valid)? And using MLE to estimate the $BF$?