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We describe a Bayesian approach to estimating quasar black hole massfunctions (BHMF) when using the broad emission lines to estimate black holemass. We show how using the broad line mass estimates in combination withstatistical techniques developed for luminosity function estimation leads tostatistically biased results. We derive the likelihood function for the BHMFbased on the broad line mass estimates, and derive the posterior distributionfor the BHMF, given the observed data. We develop our statistical approach fora flexible model where the BHMF is modelled as a mixture of Gaussian functions.Statistical inference is performed using markov chain monte carlo (MCMC)methods. Our method has the advantage that it is able to constrain the BHMFeven beyond the survey detection limits at the adopted confidence level,accounts for measurement errors and the intrinsic uncertainty in broad linemass estimates, and provides a natural way of estimating the probabilitydistribution of any quantities derived from the BHMF. We conclude by using ourmethod to estimate the local active BHMF using the z < 0.5 Bright Quasar Surveysources. At z = 0.2, the quasar BHMF falls off approximately as a power lawwith slope ~ 2 for M_{BH} > 10^8. Our analysis implies that z < 0.5 broad linequasars have a typical Eddington ratio of ~ 0.4 and a dispersion in Eddingtonratio of < 0.5 dex (abridged).

Determining Quasar Black Hole Mass Functions from their Broad Emission Lines: Application to the Bright Quasar Survey