  The majority of Quantum Random Number Generators (QRNG) are designed as
converters of a continuous quantum random variable into a discrete classical
random bit value. For the resulting random bit sequence to be minimally biased,
the conversion process demands an experimenter to fully characterize the
underlying quantum system and implement parameter estimation routines. Here we
show that conventional approaches to parameter estimation (such as e.g. {\it
Maximum Likelihood Estimation}) used on a finite QRNG data sample without
caution may introduce binning bias and lead to overestimation of the randomness
of the QRNG output. To bypass these complications, we develop an alternative
conversion approach based on the Bayesian statistical inference method. We
illustrate our approach using experimental data from a time-of-arrival QRNG and
numerically simulated data from a vacuum homodyning QRNG. Side-by-side
comparison with the conventional conversion technique shows that our method
provides an automatic on-line bias control and naturally bounds the best
achievable QRNG bit rate for a given measurement record.
