$$\begin{aligned}
\mathbb{E}(\eta_i^T \Delta Z_i | X_i, Z_i ) =  \mathbb{E}(\eta_i^T  | X_i )\Delta Z_i =  X_i^T (A^T \Delta) Z_i ,\end{aligned}$$ where $(A^T \Delta)$ is a $p \times q$ matrix of pairwise interactions between exposures and covariates. In the sequel, we focus our development on model $(1)$ for ease in exposition, but all of the details can be easily modified to pertain to model $(2)$.

Priors and MCMC Algorithm
-------------------------

In this section we define the priors for $(\omega,\Omega,\Lambda,\Psi,\sigma^2)$, briefly describe the computational challenges given by model $(1)$ and summarize our Markov Chain Monte Carlo sampler in *Algorithm 1*. We choose an Inverse-Gamma distribution with parameters $(\frac{1}{2},\frac{1}{2})$ for $\sigma^2$ and $\sigma_j^2$ for $j = 1,\cdots,p$. The elements of $\omega$ and $\Omega$ are given independent Gaussian priors. For $\Lambda = \{ \lambda_{i,j} \}$, a typical choice to attain identifiability requires $\lambda_{i,j} = 0$ for $j > i$ and $\lambda_{j,j} > 0$ for $j = 1,\cdots,k$ [@geweke1996measuring]. However, some Bayesian applications, like covariance estimation [@Bhattacharya2013], do not require identifiability of $\Lambda$. The same holds for inference on induced main effects and interactions for model $(1)$. Notice that model $(1)$ is invariant to rotations: $$\begin{aligned}
& y_i = \eta_i^T P P^T \omega + \eta_i^T 