Estimation of finite mixture models when the mixing distribution support is
unknown is an important and challenging problem. In this paper, a new approach
is given based on the recently proposed predictive recursion marginal
likelihood (PRML) method. By taking a sufficiently fine grid as a set of
candidate support points, one may treat the support itself as an unknown
parameter to be estimated. The PRML approach asymptotically integrates out the
mixing distribution itself, leaving an approximate marginal likelihood for the
support, which can be used for estimation. We employ a computationally
efficient version of simulated annealing for the large-scale combinatorial
optimization problem. Theory is given which shows that the PRML estimate will
asymptotically identify the best mixture model supported on a subset of the
candidate grid, where "best" is measured with respect to the Kullback-Leibler
divergence on the mixture scale. Real and simulated data examples show that the
PRML method compares favorably to existing Bayesian and non-Bayesian methods in
terms of mixture density estimation accuracy and model parsimony.