  Finite mixture models are an important tool in the statistical analysis of
data, for example in data clustering. The optimal parameters of a mixture model
are usually computed by maximizing the log-likelihood functional via the
Expectation-Maximization algorithm. We propose an alternative approach based on
the theory of Mean Field Games, a class of differential games with an infinite
number of agents. We show that the solution of a finite state space
multi-population Mean Field Games system characterizes the critical points of
the log-likelihood functional for a Bernoulli mixture. The approach is then
generalized to mixture models of categorical distributions. Hence, the Mean
Field Games approach provides a method to compute the parameters of the mixture
model, and we show its application to some standard examples in cluster
analysis.
