Algorithm 2.

Simplified Shotgun Stochastic Search with Screening (S5)

Set a temperature schedule t1 > t2 > …> tL > 0

Choose an initial model k(1,1) and a set of variables after screening Sk(1,1) based on k(1,1)

For l = 1 in l = L

For i in 1, …, J − 1

Compute all π(k | y) for all k ∈ nbdscr(k(i,l))

Sample k+ and k−, from ${\mathrm{\Gamma }}_{\mathit{scr}}^{+}$ and Γ−, with probabilities proportional to $\pi {(\mathbf{k}|y)}^{1/t_l}$

Sample k(i+1,l) from {k+, k−}, with probability proportional to $\{\pi {({\mathbf{k}}^{+}|y)}^{1/t_l}$, $\pi {({\mathbf{k}}^{-}|y)}^{1/t_l}\}$

Update the set of considered variables ${\mathbf{S}}_{{\mathbf{k}}^{(i+1,l)}}$ to be the union of variables in k(i+1,l) and the top Mn variables according to $\{|r_{{\mathbf{k}}^{(i+1,l)}}^T{X}_j|:j=1,\dots ,p\}$