A Flexible Method of Estimating Luminosity Functions

TL;DR

This paper introduces a Bayesian method using MCMC for estimating luminosity functions, offering more accurate confidence intervals and the ability to infer beyond survey limits compared to traditional maximum-likelihood approaches.

Contribution

It develops a flexible Bayesian approach with Gaussian mixture models and MCMC, improving luminosity function estimation and uncertainty quantification.

Findings

Bayesian confidence intervals are valid, unlike bootstrap-based ones.

The method accurately constrains luminosity functions beyond detection limits.

It enables estimation of derived quantities like quasar density peaks.

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.