Eddington's demon: Inferring galaxy mass functions and other distributions from uncertain data

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Abstract

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 multidimensional 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 emphasize the necessity of accounting for observational uncertainties in MFs of modern galaxy surveys.

Original languageEnglish
Pages (from-to)5500-5522
Number of pages23
JournalMonthly Notices of the Royal Astronomical Society
Volume474
Issue number4
DOIs
Publication statusPublished - 1 Mar 2018

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galaxies
distribution functions
estimators
angular momentum
halos
statistics
formalism
method
distribution
modeling
code
parameter
test

Cite this

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title = "Eddington's demon: Inferring galaxy mass functions and other distributions from uncertain data",
abstract = "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 multidimensional 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 emphasize the necessity of accounting for observational uncertainties in MFs of modern galaxy surveys.",
keywords = "Galaxies: luminosity function, Mass function, Methods: statistical",
author = "D. Obreschkow and Murray, {S. G.} and Robotham, {A. S.G.} and T. Westmeier",
year = "2018",
month = "3",
day = "1",
doi = "10.1093/mnras/stx3155",
language = "English",
volume = "474",
pages = "5500--5522",
journal = "Monthly Notices of the Royal Astronomical Society",
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TY - JOUR

T1 - Eddington's demon

T2 - Inferring galaxy mass functions and other distributions from uncertain data

AU - Obreschkow, D.

AU - Murray, S. G.

AU - Robotham, A. S.G.

AU - Westmeier, T.

PY - 2018/3/1

Y1 - 2018/3/1

N2 - 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 multidimensional 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 emphasize the necessity of accounting for observational uncertainties in MFs of modern galaxy surveys.

AB - 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 multidimensional 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 emphasize the necessity of accounting for observational uncertainties in MFs of modern galaxy surveys.

KW - Galaxies: luminosity function

KW - Mass function

KW - Methods: statistical

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U2 - 10.1093/mnras/stx3155

DO - 10.1093/mnras/stx3155

M3 - Article

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JO - Monthly Notices of the Royal Astronomical Society

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