Maximum likelihood estimation in the context of an optical measurement

Maximum likelihood estimation (MLE) is a widely used statistical approach for estimating one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, we review MLE's mathematical foundations within the context of measurements of optical intensity distributions. Here, the detection of each photon is treated as a random event, the outcome being the photon's location within a pixelized array. These detected photons accumulate to form an intensity profile. We give a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement. An estimate for the parameter(s) of interest is then obtained by maximizing the likelihood function, while the FIM determines the estimate's uncertainty. These concepts are illustrated with several simple examples involving a small number of pixels, for one and two parameters, revealing interesting properties and practical considerations for optical measurements. Connections are drawn to weak (off-null) measurements.