<p>The aim of this article is to detail the Bayesian approach to estimating the parameters of the Gutenberg–Richter relation modeling the rate of occurrence of an earthquake of a given magnitude. The parameters posterior distributions are known up to a factor of proportionality. They are usually estimated relying on Monte Carlo Markov Chain (MCMC) methods. These simulation techniques can be used to generate realizations of the posterior distributions, which enables to calculate useful characteristic quantities (mean, variance, quantiles, etc.). MCMC methods require fine-tuned parameterization to ensure both independence of the realizations and generation according to the desired, imperfectly known distribution. It is difficult to validate the correct parameterization of the algorithm and it requires a full simulation-based inference procedure. This article shows how the parameters posterior distributions can be computed essentially exactly (i.e. with machine precision), through a carefully rescaled version of the likelihood. Getting such an exact posterior distribution avoids the above-mentioned problems and enables highly precise evaluations of posterior quantities such as conditional expectations. Finally, the computation of the posterior normalizing constant considerably extends the applicability of the Bayesian approach to any prior distribution, hitherto often limited to the conjugate prior distribution of the model. This article applies the methodology to the Alps in France and compares the set of Bayesian results to the frequentist results based on the maximum likelihood of the Poisson process.</p>