Algorithm 1 Double-Loop Bayesian Monte Carlo (DLBMC) for estimating EIG

1:Input: Design points $\mathit{d}$, prior $p(\mathbf{\theta })$, standard deviation of noise $\sigma$, hyperparameters $\{l_{\mathrm{in}},{({\sigma }_f)}_{\mathrm{in}}\}$ and $\{l_{\mathrm{out}},{({\sigma }_f)}_{\mathrm{out}}\}$. 2:Data preparation: Sample ${\{{\mathbf{\theta }}^{(j)}\}}_{j=1}^{n_{\mathrm{in}}}\sim p(\mathbf{\theta })$ and compute $G^{(j)}=\mathit{G}(\mathit{d},{\mathbf{\theta }}^{(j)})$ for $j=1,\dots ,n_{\mathrm{in}}$. 3:Sample ${\{{\mathit{y}}^{(i)}\}}_{i=1}^{n_{\mathrm{out}}}\sim \mathcal{U}(\min (G)-\sigma ,\max (G)+\sigma )$. 4:Compute $\{z_{\mathrm{in}},K_{\mathrm{in}}\}$ and $\{z_{\mathrm{out}},K_{\mathrm{out}}\}$. 5:for$i=1,\dots ,n_{\mathrm{out}}$do 6:    for $j=1,\dots ,n_{\mathrm{in}}$ do 7:        Compute the likelihood $f^{(i,j)}=p({\mathit{y}}^{(i)}|{\mathbf{\theta }}^{(j)},\mathit{d})$. 8:    end for 9:    Let $f^{(i)}={[f^{(i,1)},\dots ,f^{(i,n_{\mathrm{in}})}]}^T$. 10:    Compute the evidence $e^{(i)}=z_{\mathrm{in}}^T{K}_{\mathrm{in}}^{-1}{f}^{(i)}$. 11:    for $j=1,\dots ,n_{\mathrm{in}}$ do 12:        $g^{(i,j)}=[\log (f^{(i,j)})-\log (e^{(i)})]f^{(i,j)}.$ 13:    end for 14:    Let $g^{(i)}={[g^{(i,1)},\dots ,g^{(i,n_{\mathrm{in}})}]}^T$. 15:    Compute $h^{(i)}=z_{\mathrm{in}}^T{K}_{\mathrm{in}}{g}^{(i)}$. 16:end for 17:Let $h={[h^{(1)},\dots ,h^{(n_{\mathrm{out}})}]}^T$. 18:Compute $\widehat{U}(\mathit{d})=z_{\mathrm{out}}^T{K}_{\mathrm{out}}^{-1}h.$ 19:Output: the estimated EIG $\widehat{U}(\mathit{d})$.