The Monte Carlo method has proved to be very powerful to cope with global illumination problems but it remains costly in terms of sampling operations. In various applications, previous work has shown that Bayesian Monte Carlo can significantly outperform importance sampling Monte Carlo thanks to a more effective use of the prior knowledge and of the information brought by the samples set. These good results have been confirmed in the context of global illumination but strictly limited to the perfect diffuse case. Our main goal in this paper is to propose a more general Bayesian Monte Carlo solution that allows dealing with non-diffuse BRDFs thanks to a spherical Gaussian-based framework. We also propose a fast hyperparameters determination method which avoids learning the hyperparameters for each BRDF. These contributions represent two major steps towards generalizing Bayesian Monte Carlo for global illumination rendering. We show that we achieve substantial quality improvements over importance sampling at comparable computational cost.