Aiming at the difficulty of obtaining the posterior distribution of measurement model in Bayesian uncertainty evaluation, a method based on accept-reject sampling is proposed to realize Bayesian measurement uncertainty evaluation. For linear/ nonlinear measurement model, the prior information being measured is obtained by using Bayesian hypothesis or Monte Carlo method, the accepted sampling points being measured are obtained based on accept-reject sampling. Then the posterior distribution is formed based on these accepted sampling points, and the measurement uncertainty evaluation results are obtained by statistical inference. Through the two evaluation examples which come from the specification and practical measurement application, it is verified that the Bayesian uncertainty evaluation method using the accept-reject algorithm can obtain reliable evaluation results compared with traditional GUM and MCM methods, the process of obtaining the Bayesian posterior distribution is simple, and it is feasible and practical in the application of measurement uncertainty evaluation under the condition of without / with historical information.