Eddington's Demon: Inferring Galaxy Mass Functions and other
  Distributions from Uncertain Data
  We present a general modified maximum likelihood (MML) method for inferring
generative distribution functions from uncertain and biased data. The MML
estimator is identical to, but easier and many orders of magnitude faster to
compute than the solution of the exact Bayesian hierarchical modelling of all
measurement errors. As a key application, this method can accurately recover
the mass function (MF) of galaxies, while simultaneously dealing with
observational uncertainties (Eddington bias), complex selection functions and
unknown cosmic large-scale structure. The MML method is free of binning and
natively accounts for small number statistics and non-detections. Its fast
implementation in the R-package "dftools" is equally applicable to other
objects, such as haloes, groups and clusters, as well as observables other than
mass. The formalism readily extends to multi-dimensional distribution
functions, e.g. a Choloniewski function for the galaxy mass-angular momentum
distribution, also handled by dftools. The code provides uncertainties and
covariances for the fitted model parameters and approximate Bayesian evidences.
We use numerous mock surveys to illustrate and test the MML method, as well as
to emphasise the necessity of accounting for observational uncertainties in MFs
of modern galaxy surveys.
