  Nonparametric maximum likelihood (NPML) for mixture models is a technique for
estimating mixing distributions that has a long and rich history in statistics
going back to the 1950s, and is closely related to empirical Bayes methods.
Historically, NPML-based methods have been considered to be relatively
impractical because of computational and theoretical obstacles. However, recent
work focusing on approximate NPML methods suggests that these methods may have
great promise for a variety of modern applications. Building on this recent
work, a class of flexible, scalable, and easy to implement approximate NPML
methods is studied for problems with multivariate mixing distributions.
Concrete guidance on implementing these methods is provided, with theoretical
and empirical support; topics covered include identifying the support set of
the mixing distribution, and comparing algorithms (across a variety of metrics)
for solving the simple convex optimization problem at the core of the
approximate NPML problem. Additionally, three diverse real data applications
are studied to illustrate the methods' performance: (i) A baseball data
analysis (a classical example for empirical Bayes methods), (ii)
high-dimensional microarray classification, and (iii) online prediction of
blood-glucose density for diabetes patients. Among other things, the empirical
results demonstrate the relative effectiveness of using multivariate (as
opposed to univariate) mixing distributions for NPML-based approaches.
