# The Likelihood Function for Poisson Measurements

Assume you perform an experiment that returns $n$ independent not identically distributed Poisson random variables $Y^{n} \triangleq \{Y_i\}_{i=1}^{n}$ each of which has mean $\lambda_i(\theta)$, where $\theta$ is a vector of parameters which belongs to some set $\Theta \subseteq \mathbb{R}^m$ called parameter space. In this setting, $\{\lambda_i\}_{i=1}^{n}$ are called parametric models.

The likelihood function can be expressed as

Taking the natural logarithm of $P_{\theta}(Y^{n} = y^{n})$, it follows that

\begin{align} \log P_{\theta}(Y^{n} = y^{n}) = \sum_{i=1}^{n}\left(- \lambda_i(\theta) + y_i\log\lambda_i(\theta) - \log y_i !\right) \end{align}

The Maximum Likelihood Estimator can be stated as the solution of the following optimization problem

\begin{align} \arg \max_{\theta \in \Theta} \log P_{\theta}(Y^{n}) & = \arg \min_{\theta \in \Theta} - \log P_{\theta}(Y^{n} = y^{n}) \\ \arg \max_{\theta \in \Theta} \log P_{\theta}(Y^{n}) & = \arg \min_{\theta \in \Theta} \sum_{i=1}^{n}\left(\lambda_i(\theta) - y_i\log\lambda_i(\theta)\right) \end{align}

To solve $\arg \max_{\theta \in \Theta} \log P_{\theta}(Y^{n})$ is necessary to solve the following set of equations (let’s talk about 2nd order derivatives and regularization conditions later)

for $j=1, 2, ..., m$. In other words, we need to find $\theta^{*} \in \Theta$ such that the above statements are verified to be true.

We can’t get any further unless we make some assumptions about the parametric model $\lambda_i$ and the parameter space.

In some problems, such as point spread function photometry and single molecule localization microscopy, one is often interested in estimating the total light flux emitted by a star (or a molecule) and its subpixel position on a detector. Therefore, our parameter vector may be written as $\theta = (A, c_o, r_o)$.

It is also practical to assume that the rate of change of the expected number of counts on a given pixel, with respect to the total integrated flux, is proportional to the expected number of counts on that pixel . Mathematically,

for some non-zero constant $\alpha$.

Substituting this assumption in (1), it follows that

The mathematical result above tells us that the total flux is exactly estimated by our model $\lambda_i$ at the solution ${\theta^{*}}$.