The main objective of this paper is to develop a convenient Bayesian procedure that can be used to assign a univariate time series realization to one of several first order moving average sources, with unknown coefficients, that share a common unknown precision. The foundation of the proposed procedure is to develop the marginal posterior mass function of a classification vector using an approximate conditional likelihood function. A time series realization is assigned to that first order moving average process with the largest posterior probability. A comprehensive simulation study with two sources is carried out to demonstrate the performance of the proposed procedure and to check its adequacy in handling the classification problems.