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

The method of maximum likelihood estimation (MLE) is a widely used statistical approach for estimating the values of one or more unknown parameters of a probabilistic model based on observed data. In this tutorial, I briefly review the mathematical foundations of MLE, then reformulate the problem for the measurement of a spatially-varying optical intensity distribution. In this context, the detection of each individual photon is treated as a random event, the outcome being the photon's location. A typical measurement consists of many detected photons, which accumulate to form a spatial intensity profile. Here, I show a straightforward derivation for the likelihood function and Fisher information matrix (FIM) associated with a measurement of multiple photons incident on a detector comprised of a discrete array of pixels. An estimate for the parameter(s) of interest may then be obtained by maximizing the likelihood function, while the FIM determines the uncertainty of the estimate. To illustrate these concepts, several simple examples are presented for the one- and two-parameter cases, revealing many interesting properties of the MLE formalism, as well as some practical considerations for optical experiments. Throughout these examples, connections are also drawn to optical applications of quantum weak measurements, including off-null ellipsometry and scatterometry.