### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 874-895 |

Number of pages | 22 |

Journal | Astrophysical Journal |

Volume | 682 |

Issue number | 2 |

DOIs | |

State | Published - Aug 1 2008 |

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### Keywords

- Methods: data analysis
- Methods: numerical
- Methods: statistical

### ASJC Scopus subject areas

- Space and Planetary Science
- Astronomy and Astrophysics

### Cite this

*Astrophysical Journal*,

*682*(2), 874-895. https://doi.org/10.1086/589501

**A flexible method of estimating luminosity functions.** / Kelly, Brandon C.; Fan, Xiaohui; Vestergaard, Marianne.

Research output: Contribution to journal › Article

*Astrophysical Journal*, vol. 682, no. 2, pp. 874-895. https://doi.org/10.1086/589501

}

TY - JOUR

T1 - A flexible method of estimating luminosity functions

AU - Kelly, Brandon C.

AU - Fan, Xiaohui

AU - Vestergaard, Marianne

PY - 2008/8/1

Y1 - 2008/8/1

N2 - 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.

AB - 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.

KW - Methods: data analysis

KW - Methods: numerical

KW - Methods: statistical

UR - http://www.scopus.com/inward/record.url?scp=53549097958&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=53549097958&partnerID=8YFLogxK

U2 - 10.1086/589501

DO - 10.1086/589501

M3 - Article

VL - 682

SP - 874

EP - 895

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 2

ER -