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The majority of Quantum Random Number Generators (QRNG) are designed asconverters of a continuous quantum random variable into a discrete classicalrandom bit value. For the resulting random bit sequence to be minimally biased,the conversion process demands an experimenter to fully characterize theunderlying quantum system and implement parameter estimation routines. Here weshow that conventional approaches to parameter estimation (such as e.g. {\itMaximum Likelihood Estimation}) used on a finite QRNG data sample withoutcaution may introduce binning bias and lead to overestimation of the randomnessof the QRNG output. To bypass these complications, we develop an alternativeconversion approach based on the Bayesian statistical inference method. Weillustrate our approach using experimental data from a time-of-arrival QRNG andnumerically simulated data from a vacuum homodyning QRNG. Side-by-sidecomparison with the conventional conversion technique shows that our methodprovides an automatic on-line bias control and naturally bounds the bestachievable QRNG bit rate for a given measurement record.

An Observed-Data-Consistent Approach to the Assignment of Bit Values in a Quantum Random Number Generator