The frequency/magnitude distribution of earthquakes can be approximated by an exponential law whose exponent (the so-called b-value) is routinely used for probabilistic seismic hazard assessment. The b-value is commonly measured using Aki’s maximum likelihood estimation, although biases can arise from the choice of completeness magnitude (i.e. the magnitude below which the exponential law is no longer valid). In this work, we introduce the b-Bayesian method, where the full frequency-magnitude distribution of earthquakes is modelled by the product of an exponential law and a detection law. The detection law is characterized by two parameters, which we jointly estimate with the b-value within a Bayesian framework. All available data are used to recover the joint probability distribution. The b-Bayesian approach recovers temporal variations of the b-value and the detectability using a transdimensional Markov chain Monte Carlo (McMC) algorithm to explore numerous configurations of their time variations. An application to a seismic catalog of far-western Nepal shows that detectability decreases significantly during the monsoon period, while the b-value remains stable, albeit with larger uncertainties. This confirms that variations in the b-value can be estimated independently of variations in detectability (i.e. completeness). Our results are compared with those obtained using the maximum likelihood estimation, and using the b-positive approach, showing that our method avoids dependence on arbitrary choices such as window length or completeness thresholds.