($\tau_j$) and local ($\phi_{jh}$) scales. This induces a prior with concentration at zero, to strongly shrink small signals, and heavy-tails, to avoid over-shrinking large signals. The DL prior induces near sparsity row-wise in the matrix $\Lambda$, as it is reasonable to assume that each variable loads on few factors.

In *Section 2.4*, we describe how the above prior specification induces an appealing shrinkage prior on the main effects and interactions, and discuss hyperparameter choice. In practice, we recommend the rule of thumb that chooses $k$ such that $\frac{\sum_{j = 1}^k v_j}{\sum_{j = 1}^p v_j} > 0.9$, where $v_j$ is the $j^{th}$ largest singular value of the correlation matrix of $X$. *Proposition 2* in *Section 3* provides theoretical justification for this criterion. As an alternative to row-wise shrinkage, we could have instead used column-wise shrinkage as advocated in [@Bhattacharya2015] and [@legramanti2019bayesian]. Although such approaches can be effective in choosing the number of factors, we found in our simulations that they can lead to over-shrinkage of the estimated main effects and interactions.

The inclusion of pairwise interactions among the factors in the regression of the outcome $y_i$ rules out using a simple data augmentation Gibbs sampler, as in [@Mike03], [@Bhattacharya2013]. The log 