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.
"Extreme deconvolution: Inferring complete distribution functions from noisy, heterogeneous and incomplete observations." Ann. Appl. Stat. 5 (2B) 1657 - 1677, June 2011. https://doi.org/10.1214/10-AOAS439