  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.
