We propose a Bayesian method to extract the diffusivity of biomolecules evolving freely or inside membrane microdomains. This approach assumes a model of motion for the particle considered, namely free Brownian motion or confined diffusion. In each framework, a systematic Bayesian scheme is provided for estimating the diffusivity. We show that this method reaches the best performances theoretically achievable. Its efficiency overcomes that of widely used methods based on the analysis of the mean-square displacement. The approach presented here also gives direct access to the uncertainty on the estimation of the diffusivity and predicts the number of steps of the trajectory necessary to achieve any desired precision. Its robustness with respect to noise on the position of the biomolecule is also investigated.