Model identification is the first and most important stage when analyzing a time series. As a result of analytical complexity, very little has been done from a Bayesian viewpoint in order to identify the orders of ARMA models. Some analytical and numerical identification Techniques have been introduced by Monahan (1983), Broemeling and Shaarawy (1987) and Ali (2003). However, the performance of Monahan (1983) and Ali (2003) techniques have not been studied yet through an effectiveness study. This article has three different objectives. The first one is to carry out a simulation study to assess the performance and efficiency of Monahan's technique. The second objective is to carry out a simulation study to test the adequacy of Ali's technique in handling the Identification problem of ARMA processes. The third objective is to compare among the three Bayesian Identification Techniques   through a comprehensive simulation study. In addition, the results of the three Bayesian Identification Techniques   are compared with the well-known non-Bayesian automatic technique, AIC. The numerical results support the adequacy of the Bayesian techniques in solving the identification problems of autoregressive moving average processes.