distributions with different variances, and that we place a higher mixture weight on the normal distribution concentrated around zero as we increase the order of interactions. This is because higher order interactions contain products of the elements of $A$, previously defined in *Proposition 1*, and the elements of $A$ are affected by the DL prior shrinkage, since $A$ is a function of $\Lambda$. Also, notice that the priors have higher variance as we increase the number of latent factors $k$.

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In environmental epidemiology, it is common to have prior knowledge of groups of exposures that are highly correlated and it is natural to include such information in the specification of $\Lambda$. One possibility is to impose a block sparsity structure in which each group of chemicals is restricted to load on the same factor. Then, cross group dependence is allowed including additional factors and endowing the factor loadings with a DL prior. Suppose that the variables in $X$ can be divided in $l$ groups: $S_1, S_2, \ldots S_l$ of dimensions $p_1,p_2,\ldots,p_l$, where $l < k$ and $p = \sum_{r = 1}^l p_r$. Letting $\Lambda = [\Lambda^B \Lambda']$, where $\Lambda^B$ is $p \times l$, we can assign a block sparsity 