together with enforcing $\Lambda$ to be orthogonal, we have that $\beta_X  = \Lambda \omega$. It follows that $\beta_X$ has the *generalised singular g-prior* (or *gsg*-prior) distribution defined by [@Mike03], whose density is proportional to $ exp(-\frac{1}{2} \beta_X^T \Lambda^T \Xi^{-1} \Lambda \beta)$.

Now, consider the extension presented in the previous section, where we include powers of the factors in the regression of $y_i$. In *Figure 1*, we show the induced marginal priors for main effects, pairwise interactions and $3^{rd}$ order interactions when $p = 20$ and $k = 5,10$ when $\omega$ and $\Omega$ are given $N(0,1)$ priors element-wise. Increasing (or decreasing) the variance of the priors on $\omega$ and $\Omega$ will directly increase (or decrease) the variance of the induced main effects and pairwise interactions, as $\beta_X$ and $\Omega_X$ are linear functions of $\omega$ and $\Omega$ respectively. For a fixed $k$, there is increasing shrinkage towards zero with higher orders of interaction. However, we avoid assuming exact sparsity corresponding to zero values of the coefficients, a standard assumption of other methods. Although most of the mass is concentrated around zero, the distributions have heavy tails. We can indeed notice that the form of the priors resembles a mixture of two normal 