A Flexible Method of Estimating Luminosity Functions

Abstract

We describe a Bayesian approach to estimating luminosity functions. We derive the likelihood function and posterior probability distribution for the luminosity function, given the observed data, and we compare the Bayesian approach with maximum-likelihood by simulating sources from a Schechter function. For our simulations confidence intervals derived from bootstrapping the maximum-likelihood estimate can be too narrow, while confidence intervals derived from the Bayesian approach are valid. We develop our statistical approach for a flexible model where the luminosity function is modeled as a mixture of Gaussian functions. Statistical inference is performed using Markov chain Monte Carlo (MCMC) methods, and we describe a Metropolis-Hastings algorithm to perform the MCMC. The MCMC simulates random draws from the probability distribution of the luminosity function parameters, given the data, and we use a simulated data set to show how these random draws may be used to estimate the probability distribution for the luminosity function. In addition, we show how the MCMC output may be used to estimate the probability distribution of any quantities derived from the luminosity function, such as the peak in the space density of quasars. The Bayesian method we develop has the advantage that it is able to place accurate constraints on the luminosity function even beyond the survey detection limits, and that it provides a natural way of estimating the probability distribution of any quantities derived from the luminosity function, including those that rely on information beyond the survey detection limits.

26 pages, 13 figures, emulateapj format, accepted by ApJ. An IDL computer routine is available from B. Kelly for implementing our technique of estimating luminosity functions

References in corpus (52)

• First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters
• The Demography of Massive Dark Objects in Galaxy Centres
• The broken hierarchy of galaxy formation
• A Unified, Merger-Driven Model for the Origin of Starbursts, Quasars, the Cosmic X-Ray Background, Supermassive Black Holes and Galaxy Spheroids
• Local Supermassive Black Holes, Relics of Active Galactic Nuclei and the X-ray Background
• Some Aspects of Measurement Error in Linear Regression of Astronomical Data
• The Galaxy Luminosity Function and Luminosity Density at Redshift z=0.1
• An Observational Determination of the Bolometric Quasar Luminosity Function
• Galaxy Luminosity Functions to z~1: DEEP2 vs. COMBO-17 and Implications for Red Galaxy Formation
• Cosmological Evolution of the Hard X-ray AGN Luminosity Function and the Origin of the Hard X-ray Background
• Observational constraints on growth of massive black holes
• The SDSS Quasar Survey: Quasar Luminosity Function from Data Release Three
• The 2dF QSO Redshift Survey - XII. The spectroscopic catalogue and luminosity function
• A Survey of z>5.7 Quasars in the Sloan Digital Sky Survey IV: Discovery of Seven Additional Quasars
• Self-Regulated Growth of Supermassive Black Holes in Galaxies as the Origin of the Optical and X-ray Luminosity Functions of Quasars
• The Cosmic Evolution of Hard X-ray Selected Active Galactic Nuclei
• The HELLAS2XMM survey. VII. The hard X-ray luminosity function of AGN up to z=4: more absorbed AGN at low luminosities and high redshifts
• The Masses of Nuclear Black Holes in Luminous Elliptical Galaxies and Implications for the Space Density of the Most Massive Black Holes
• High-Redshift Quasars Found in Sloan Digital Sky Survey Commissioning Data IV: Luminosity Function from the Fall Equatorial Stripe Sampl
• Radio sources in the 6dFGS: Local luminosity functions at 1.4 GHz for star-forming galaxies and radio-loud AGN
• On the Relationship Between Radio Emission and Black Hole Mass in Galactic Nuclei
• The Evolving Luminosity Function of Red Galaxies
• The radio luminosity function from the low-frequency 3CRR, 6CE & 7CRS complete samples
• The Sloan Digital Sky Survey Quasar Catalog III. Third Data Release
• The evolution of faint AGN between z~1 and z~5 from the COMBO-17 survey
• Luminosity Function of Morphologically Classified Galaxies in the Sloan Digital Sky Survey
• Active Galactic Nuclei in the Sloan Digital Sky Survey: II. Emission-Line Luminosity Function
• The evolution of the near-IR galaxy Luminosity Function and colour bimodality up to z ~ 2 from the UKIDSS Ultra Deep Survey Early Data Release
• The anti-hierarchical growth of supermassive black holes
• The Changing AGN Population
• The Luminosity Function of Void Galaxies in the Sloan Digital Sky Survey
• Luminosity functions for galaxies and quasars in the Spitzer Wide-Area Infrared Extra-galactic (SWIRE) Legacy survey
• An improved method of constructing binned luminosity functions
• A Transition in the Accretion Properties of Radio-Loud Active Nuclei
• The rest-frame optical luminosity functions of galaxies at 2<z<3.5
• The Evolution of the Optical and Near-Infrared Galaxy Luminosity Functions and Luminosity Densities to z~2
• The Physical and Photometric Properties of High-Redshift Galaxies in Cosmological Hydrodynamic Simulations
• Radio Sources in Galaxy Clusters: Radial Distribution, and 1.4 GHz and K-band Bivariate Luminosity Function
• The LBDS Hercules sample of mJy radio sources at 1.4 GHz - II. Redshift distribution, radio luminosity function, and the high-redshift cut-off
• How Much Mass do Supermassive Black Holes Eat in their Old Age?
• A Spectroscopic Survey of Faint Quasars in the SDSS Deep Stripe: I. Preliminary Results from the Co-added Catalog
• The Ultraviolet Luminosity Function of GALEX Galaxies at Photometric Redshifts Between 0.07 and 0.25
• Active Galactic Nuclei in the mid-IR. Evolution and Contribution to the Cosmic Infrared Background
• A simple test for the existence of two accretion modes in Active Galactic Nuclei
• The Evolution and Luminosity Function of Quasars from Complete Optical Surveys
• The Far-Infrared Luminosity Function from GOODS-N: Constraining the Evolution of Infrared Galaxies for z \leq 1
• Probing the Low-Luminosity XLF in Normal Elliptical Galaxies
• A Statistical Method for Estimating Luminosity Functions using Truncated Data
• Assessing the Predictive Power of Galaxy Formation Models: A Comparison of Predicted and Observed Rest-Frame Optical Luminosity Functions at 2.0<z<3.3
• The VIMOS VLT Deep Survey:The ultraviolet galaxy luminosity function and luminosity density at 3<z<4
• The cluster galaxy luminosity function at $z=0.3$: a recent origin for the faint-end upturn ?
• Constraints on radiatively inefficient accretion history from Eddington ratio distribution of active galactic nuclei

Cited by in corpus (41)

• Mass Functions of the Active Black Holes in Distant Quasars from the Large Bright Quasar Survey, the Bright Quasar Survey, and the Color-Selected Sample of the SDSS Fall Equatorial Stripe
• Kinematics in Young Star Clusters and Associations with Gaia DR2
• Obscuration-dependent evolution of Active Galactic Nuclei
• The z=5 Quasar Luminosity Function from SDSS Stripe 82
• On the normalised FRB luminosity function
• The Luminosity Function at z~8 from 97 Y-band dropouts: Inferences About Reionization
• X-rays across the galaxy population I: tracing the main sequence of star formation
• Correcting the z~8 Galaxy Luminosity Function for Gravitational Lensing Magnification Bias
• X-ray binary formation in low-metallicity blue compact dwarf galaxies
• The Evolution of the Lyman-Alpha Luminosity Function During Reionization
• Shedding Light on the Galaxy Luminosity Function
• Mass Functions of Supermassive Black Holes Across Cosmic Time
• The 5 - 10 keV AGN luminosity function at 0.01<z<4.0
• Galaxy gas as obscurer: I. GRBs x-ray galaxies and find a N_H ~ M* relation
• Determining Quasar Black Hole Mass Functions from their Broad Emission Lines: Application to the Bright Quasar Survey
• The HerMES sub-millimetre local and low-redshift luminosity functions
• Probabilistic Catalogs for Crowded Stellar Fields
• Evolution of the far-infrared luminosity functions in the Spitzer Wide-area Infrared Extragalactic Legacy Survey
• A Monte Carlo Approach to Evolution of the Far-Infrared Luminosity Function with BLAST
• Probing Gas Motions in the Intra-Cluster Medium: A Mixture Model Approach
• A graphical analysis of the systematic error of classical binned methods in constructing luminosity functions
• Biases and Cosmic Variance in Molecular Gas Abundance Measurements
• The Effects of X-Ray and UV Background Radiation on the Low-Mass Slope of the Galaxy Mass Function
• Dynamical Models for NGC 6503 using a Markov Chain Monte Carlo Technique
• A mixture evolution scenario of the AGN radio luminosity function
• Introducing the Photometric Maximum Likelihood Method: Galaxy Luminosity Functions at z<1.2 in MUSYC-ECDFS
• A flexible method for estimating luminosity functions via Kernel Density Estimation
• Two per cent measurement of $H_0$ from Cepheids alone
• Effect of spectral index distribution on estimating the AGN radio luminosity function
• The Velocity Field Olympics: Assessing velocity field reconstructions with direct distance tracers
• No evidence for local $H_0$ anisotropy from Tully--Fisher or supernova distances
• The subtle statistics of the distance ladder: On the distance prior and selection effects
• An approach to robust Bayesian regression in astronomy
• Completeness III: identifying characteristic systematics and evolution in galaxy redshift surveys
• Cross-disciplinary research in astronomy
• GPU-Accelerated Hierarchical Bayesian Inference with Application to Modeling Cosmic Populations: CUDAHM
• Quasar Mass Functions Across Cosmic Time
• A flexible method for estimating luminosity functions via Kernel Density Estimation -- II. Generalization and Python implementation
• $S_8$ from peculiar velocities: agreement with Planck for Tully--Fisher and supernovae, tension for the fundamental plane
• Bayesian luminosity function estimation in multi-depth datasets with selection effects: A case study for $3<z<5$ Lyman $\alpha$ emitters
• BASS XXX: Distribution Functions of DR2 Eddington-ratios, Black Hole Masses, and X-ray Luminosities