  Predictive recursion (PR) is a fast, recursive algorithm that gives a smooth
estimate of the mixing distribution under the general mixture model. However,
the PR algorithm requires evaluation of a normalizing constant at each
iteration. When the support of the mixing distribution is of relatively low
dimension, this is not a problem since quadrature methods can be used and are
very efficient. But when the support is of higher dimension, quadrature methods
are inefficient and there is no obvious Monte Carlo-based alternative. In this
paper, we propose a new strategy, which we refer to as a PRticle filter,
wherein we augment the basic PR algorithm with a filtering mechanism that
adaptively reweights an initial set of particles along the updating sequence
which are used to obtain Monte Carlo approximations of the normalizing
constants. Convergence properties of the PRticle filter approximation are
established and its empirical accuracy is demonstrated with simulation studies
and a marked spatial point process data analysis.
