  We describe a Bayesian approach to estimating quasar black hole mass
functions (BHMF) when 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 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 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 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} > 10^8. Our analysis implies that z < 0.5 broad line
quasars have a typical Eddington ratio of ~ 0.4 and a dispersion in Eddington
ratio of < 0.5 dex (abridged).
