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In this short paper, we discuss a novel way of constructing prior distributions for correlation matrices and an associated approach to inference. We construct a prior penalizing large correlations, which we incorporate into an oblique factor model and a Candecomp/Parafac model for three-way data. We argue that this choice of prior for the factor correlation matrix, combined with a shrinkage prior for elements of the factor loadings matrix, leads to interpretable solutions. At the meeting we will demonstrate this through applications to real data.