evaluate the performance on a test dataset of $500$ units computing predictive mean square error, mean square error for main effects, Frobenious norm (FR) for interaction effects, and percentage of true positives (TP) and true negatives (TN) for main effects and interactions. The percentage of TP and TN main effects is defined as follows: $$\begin{aligned}
& \text{TP}(\text{main effects}) = \frac{1}{p} \sum_{j = 1}^p \mathbbm{1}(\hat{\beta_j} \neq 0 ,\beta_{0j} \neq 0 , sign(\hat{\beta_j}) = sign(\beta_{0j}) ) \\
& \text{TN}(\text{main effects}) = \frac{1}{p} \sum_{j = 1}^p \mathbbm{1}(\hat{\beta}_j = 0,\beta_{0j} = 0   ),\end{aligned}$$ where $\hat{\beta_j}$ is the estimated main effect for feature $j$ and $\beta_{0j}$ is the true coefficient. FIN is the only method reporting uncertainty quantification and we set $\hat{\beta_j} = 0$ whenever zero is included in the $95 \%$ credible interval. We equivalently define the percentage of true positive and true negative interactions.

The MCMC algorithm was run for $5000$ iterations with a burn-in of $4000$. We observed good mixing. In particular, the Effective Sample Size (ESS) was always greater than $900$ across our simulations, both for main effects and interactions. We set the hyperparameter $a$ of the Dirichlet-Laplace prior equal to $1/2$. We obtained similar results for $a$ in the interval 