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Expressing data as linear functions of a small number of unknown variables is a useful approach employed by several classical data analysis methods, e.g., factor analysis, principal component analysis, or latent semantic indexing. These models represent the data using the product of two factors. In practice, one important concern is how to link the learned factors to relevant quantities in the context of the application. To this end, various specialized forms of the factors have been proposed to improve interpretability. Toward developing a unified view and clarifying the statistical significance of the specialized factors, we propose a Bayesian model family. We employ exponential family distributions to specify various types of factors, which provide a unified probabilistic formulation. A Gibbs sampling procedure is constructed as a general computation routine. We verify the model by experiments, in which the proposed model is shown to be effective in both emulating existing models and motivating new model designs for particular problem settings.