Determining quasar black hole mass functions from their broad emission lines: Application to the bright Quasar Survey

We describe a Bayesian approach to estimating quasar black hole mass functions (BHMF) using the broad emission lines to estimate black hole mass. We show how using the broad-line mass estimates in combination with statistical techniques developed for luminosity function estimation (e.g., the 1/V _a correction) leads to statistically biased results. We derive the likelihood function for the BHMF based on the broad-line mass estimates, and derive the posterior distribution for the BHMF, given the observed data. We develop our statistical approach for a flexible model where the BHMF 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 BHMF 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 BHMF. In addition, we show how the MCMC output may be used to estimate the probability distribution of any quantities derived from the BHMF, such as the peak in the space density of quasars. Our method has the advantage that it is able to constrain the BHMF even beyond the survey detection limits at the adopted confidence level, accounts for measurement errors and the intrinsic uncertainty in broad-line mass estimates, and provides a natural way of estimating the probability distribution of any quantities derived from the BHMF. We conclude by using our method to estimate the local active BHMF using the z < 0.5 Bright Quasar Survey sources. At z ∼ 0.2, the quasar BHMF falls off approximately as a power law with slope ∼2 for M_BH ≳ 10⁸ M_⊙. Our analysis implies that at a given M_BH, z < 0.5 broad-line quasars have a typical Eddington ratio of ∼0.4 and a dispersion in Eddington ratio of ≲0.5 dex.