structure to $\Lambda^B$: $$\begin{aligned}
    & \lambda^B_{p_1 +1, 1} = \ldots = \lambda^B_{p, 1} = 0 \\ 
    & \lambda^B_{1, 2} = \ldots = \lambda^B_{p_1, 2} = \lambda^B_{p_1 + p_2 + 1, 2} = \ldots =  \lambda^B_{p, 2} = 0 \\
   & \cdots \\
    & \lambda^B_{1, l} = \ldots = \lambda^B_{p - p_l, l} = 0 \end{aligned}$$

In the *Supplementary Material* we show the effect of the block sparsity structure on the a priori induced groupings of main effects and interactions when $l = k$, so that $\Lambda = \Lambda^B$.

Properties of the Model
=======================

In this section we prove that the posterior distribution of $\Theta = (\omega,\Omega,\sigma^2,\Lambda,\Psi)$ is weakly consistent for a broad set of models. Let $KL(\Theta_0,\Theta)$ denote the Kullback-Leibler divergence between $p(X,y |  \Theta_0) $ and $p(X,y|\Theta)$, where $$\begin{aligned}
p(X,y|\Theta_0) =  \int p(X| \Lambda_0,\Psi_0,\eta)p(y|\omega_0,\Omega_0,\sigma_0^2,\eta) p(\eta) d\eta .\end{aligned}$$ We will assume that $p(X,y|\Theta_0) $ represents the true data-generating model. This assumption is not as restrictive as it may initially seem. The model is flexible enough to always characterize and model quadratic regression in the response component, while accurately approximating any covariance structure in the predictor component. In fact it always holds that: $$\begin{aligned}
& \mathbb{E}(y_i| X_i) = \beta_0 X_i + 