Extreme deconvolution: Inferring complete distribution functions from
  noisy, heterogeneous and incomplete observations
  We generalize the well-known mixtures of Gaussians approach to density
estimation and the accompanying Expectation--Maximization technique for finding
the maximum likelihood parameters of the mixture to the case where each data
point carries an individual $d$-dimensional uncertainty covariance and has
unique missing data properties. This algorithm reconstructs the
error-deconvolved or "underlying" distribution function common to all samples,
even when the individual data points are samples from different distributions,
obtained by convolving the underlying distribution with the heteroskedastic
uncertainty distribution of the data point and projecting out the missing data
directions. We show how this basic algorithm can be extended with conjugate
priors on all of the model parameters and a "split-and-merge" procedure
designed to avoid local maxima of the likelihood. We demonstrate the full
method by applying it to the problem of inferring the three-dimensional
velocity distribution of stars near the Sun from noisy two-dimensional,
transverse velocity measurements from the Hipparcos satellite.
