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Zé Vinícius

Intern at NASA Ames Research Center Kepler/K2 Mission

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Assume you perform an experiment that returns independent not identically distributed Poisson random variables each of which has mean , where is a vector of parameters which belongs to some set called parameter space. In this setting, are called parametric models.

The likelihood function can be expressed as

Taking the natural logarithm of , 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 is necessary to solve the following set of equations (let’s talk about 2nd order derivatives and regularization conditions later)

for . In other words, we need to find 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 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 .

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 .

Substituting this assumption in (1), it follows that

The mathematical result above tells us that the total flux is exactly estimated by our model at the solution .