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The method of maximum likelihood estimation (MLE) is a widely usedstatistical approach for estimating the values of one or more unknownparameters of a probabilistic model based on observed data. In this tutorial, Ibriefly review the mathematical foundations of MLE, then reformulate theproblem for the measurement of a spatially-varying optical intensitydistribution. In this context, the detection of each individual photon istreated as a random event, the outcome being the photon's location. A typicalmeasurement consists of many detected photons, which accumulate to form aspatial intensity profile. Here, I show a straightforward derivation for thelikelihood function and Fisher information matrix (FIM) associated with ameasurement of multiple photons incident on a detector comprised of a discretearray of pixels. An estimate for the parameter(s) of interest may then beobtained by maximizing the likelihood function, while the FIM determines theuncertainty of the estimate. To illustrate these concepts, several simpleexamples are presented for the one- and two-parameter cases, revealing manyinteresting properties of the MLE formalism, as well as some practicalconsiderations for optical experiments. Throughout these examples, connectionsare also drawn to optical applications of quantum weak measurements, includingoff-null ellipsometry and scatterometry.

Tutorial: Maximum likelihood estimation in the context of an optical measurement