Approximate nonparametric maximum likelihood for mixture models: A convex optimization approach to fitting arbitrary multivariate mixing distributions

Nonparametric maximum likelihood (NPML) for mixture models is a technique for estimating mixing distributions, which has a long and rich history in statistics going back to the 1950s (Kiefer and Wolfowitz, 1956; Robbins, 1950). However, NPMLbased methods have been considered to be relatively impractical because of computational and theoretical obstacles. Recent work focusing on approximate NPML methods and leveraging modern computing power suggests, on the other hand, that these methods may have great promise for many interesting applications. Most of this recent work has focused on specific examples involving relatively simple statistical models and univariate mixing distributions. In this paper, we propose a general approach to fitting arbitrary multivariate mixing distributions with NPML-based methods via convex optimization. The proposed methods are highly flexible and easy to implement. We illustrate their performance in several applications involving estimation and classification.