KL((\Lambda_0,\Psi_0);(\Lambda^*,\Psi^*)) \le \sum_{j = k + 1}^{k_0} \frac{ v_j}{s_0 } ,\end{aligned}$$ where $v_j$ is the $j^{th}$ largest singular value of $\Lambda_0 \Lambda_0 ^T$.

Unsurprisingly, the bound of *Proposition 2* resembles the Eckart-Young theorem for low-rank approximation based on the Singular Value Decomposition of a matrix. The Eckart-Young theorem states that the rank $k$ approximation $\hat{\Omega}$ of a matrix $\Omega$ minimizing the Frobenoius norm is such that $||\hat{\Omega} - \Omega||_F  = \sqrt{ \sum_{j = k+1}^p v_j^2} $. In a similar fashion as Principal Component Analysis and Factor Analysis, we can inspect the singular values of the correlation matrix of the regressors in order to choose the number of factors to include in the model, and thanks to *Proposition 2* we know how far the posterior distribution will concentrate from the truth.

The next proposition provides a sufficient condition in order to achieve posterior consistency when $k \ge k_0$. Notice that we achieve posterior consistency on the induced main effects and pairwise interactions.

Fix $\Theta_0 = (\omega_0,\Omega_0,\sigma_{0}^2,\Lambda_0,\Psi_0,k_0)$. Whenever $k \ge k_0$, for any $\delta > 0$ there exists an $\epsilon > 0$ such that: $$\begin{aligned}
\{\Theta: d_{\infty}(\Theta_0,\Theta)< \delta \} \subset \{\Theta: KL(\Theta_0,\Theta)< \epsilon \}\end{aligned}$$ where $d_\infty$ is the sup-norm.

One can easily 