25$. These two simulation scenarios are the most similar to the environmental epidemiology data analysis in *Section 5*. The complexity gains of FIN with respect to a Bayesian linear models with interactions is analyzed in the *Supplementary Material*.

For each scenario, we generate the continuous outcome according to a linear regression with pairwise interactions: $$\begin{aligned}
y_i = X_i^T \beta_0  + X_i^T \Omega_0 X_i + \epsilon_i,\end{aligned}$$ where half of the main effects are different from zero and $\epsilon_i \sim N(0,1)$ for $i = 1,\cdots,500$. We distinguish between a sparse matrix of pairwise interactions $\Omega_0$, with only $5\%$ non-zero interactions, or dense, where $20\%$ of the elements are different from zero.

For each value of $p$ we have six simulation scenarios: factor, linear or independent combined with sparse or dense pairwise interactions. We generate the non-zero main effects and interaction coefficients from a Uniform distribution in the interval $(-1,-0.5) \bigcup (0.5,1)$ such that the regression equation follows the strong heredity constraint. Strong heredity allows an interaction between two variables to be included in the model only if the main effects are. This is done to favor RAMP, Family and HierNet, which assume the heredity condition. We repeat the simulations $50$ times and 