A Bayesian approach for classification of Markov sources is developed and studied. Each of M sources is described by a continuous-time, discrete-state Markov chain. All states and times of transitions between states can be observed perfectly but the transition rate matrices which establish the parameters of the sources are not known a priori. A Bayesian training algorithm using a fixed amount of memory digests the training samples that consists of a member function from each chain. This leads to an iterative and computationally simple classification and training algorithms. It is also shown the convergency of the parameter posterior density to the true value of the unknown parameters. Finally, the algorithms established are applied for classification of Electroencephalograms (EEG). Details of the initial data reduction, primitive feature selection which translate an EEG record into Markov states and the lengths of the sojourns in each state are explained. The experimental results on the real data demonstrates that this approach can provide substantial help to the clinicans and researchers who required EEG analysis.