when $q$ is odd and on the bottom summation for $f = \frac{q}{2}$ when $q$ is even. Finally notice that with $Q k$ parameters we manage to estimate $\sum_{q =  0}^Q {p \choose q}$ parameters thanks to the low dimensional factor structure in the covariates.

Induced Priors
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In this section, we show the behavior of the induced priors on the main effects and pairwise interaction coefficients under model $(1)$ using simulated examples, and we show the induced grouping of coefficients when we have prior information on the covariance structure of $X$. We endow $\omega$ with a normal prior having zero mean and covariance equal to $\Xi$, where $\Xi$ is a diagonal matrix. Then, conditional on $\Lambda$ and $\Psi$, the induced prior on $\beta_X$ is also Normal with mean $0$ and covariance equal to $A^T A$. Recall from *Proposition 1* that the induced main effect coefficients are equal to $\beta_X^T = \omega^T (\Lambda^T \Psi^{-1} \Lambda)^{-1} \Lambda^T \Psi^{-1} $. This expression is equivalent to [@Mike03] and we can similarly characterize the limiting case of $\Psi \rightarrow 0$, i.e. when the factors explain all of the variability in the matrix of regressors $X_i$. Let $\Psi = s I$ and $s \rightarrow 0$, 