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Early diagnosis and assessment of fatal diseases and acute infections on chest X-ray (CXR) imaging may have important therapeutic implications and reduce mortality. In fact, many respiratory diseases have a serious impact on the health and lives of people. However, certain types of infections may include high variations in terms of contrast, size and shape which impose a real challenge on classification process. This paper introduces a new statistical framework to discriminate patients who are either negative or positive for certain kinds of virus and pneumonia. We tackle the current problem via a fully Bayesian approach based on a flexible statistical model named shifted-scaled Dirichlet mixture models (SSDMM). This mixture model is encouraged by its effectiveness and robustness recently obtained in various image processing applications. Unlike frequentist learning methods, our developed Bayesian framework has the advantage of taking into account the uncertainty to accurately estimate the model parameters as well as the ability to solve the problem of overfitting. We investigate here a Markov Chain Monte Carlo (MCMC) estimator, which is a computer-driven sampling method, for learning the developed model. The current work shows excellent results when dealing with the challenging problem of biomedical image classification. Indeed, extensive experiments have been carried out on real datasets and the results prove the merits of our Bayesian framework.