  In this paper, we outline the use of Mixture Models in density estimation of
large astronomical databases. This method of density estimation has been known
in Statistics for some time but has not been implemented because of the large
computational cost. Herein, we detail an implementation of the Mixture Model
density estimation based on multi-resolutional KD-trees which makes this
statistical technique into a computationally tractable problem. We provide the
theoretical and experimental background for using a mixture model of Gaussians
based on the Expectation Maximization (EM) Algorithm. Applying these analyses
to simulated data sets we show that the EM algorithm - using the AIC penalized
likelihood to score the fit - out-performs the best kernel density estimate of
the distribution while requiring no ``fine--tuning'' of the input algorithm
parameters. We find that EM can accurately recover the underlying density
distribution from point processes thus providing an efficient adaptive
smoothing method for astronomical source catalogs. To demonstrate the general
application of this statistic to astrophysical problems we consider two cases
of density estimation: the clustering of galaxies in redshift space and the
clustering of stars in color space. From these data we show that EM provides an
adaptive smoothing of the distribution of galaxies in redshift space
(describing accurately both the small and large-scale features within the data)
and a means of identifying outliers in multi-dimensional color-color space
(e.g. for the identification of high redshift QSOs). Automated tools such as
those based on the EM algorithm will be needed in the analysis of the next
generation of astronomical catalogs (2MASS, FIRST, PLANCK, SDSS) and ultimately
in in the development of the National Virtual Observatory.
