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

Approximate nonparametric maximum likelihood inference for mixture models via convex optimization