Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In this work, we propose a method for estimating the Hurst index, or memory parameter, of a stationary process with long memory in a Bayesian fashion. Such approach provides an approximation for the posterior distribution for the memory parameter and it is based on a simple application of the so-called approximate Bayesian computation (ABC), also known as likelihood-free method. Some popular existing estimators are reviewed and compared to this method for the fractional Brownian motion, for a long-range binary process and for the Rosenblatt process. The performance of our proposal is remarkably efficient.46212191237Sao Paulo Research Foundation [2013/07699-0]CNPq at the University of Sao Paulo [141048/2013-1]Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)