$q^*=J_{\delta}(y,z^{*})$. 3. Generate $\theta^{\#}:= \texttt{AMRW}(\theta^*,\Sigma_j)$. Generate $X^{\#}$’s from $p(X|\theta^{\#})$ and corresponding $z^{\#}$. Compute $q^{\#}=J_{\delta}(y,z^{\#})$. 4. Generate $\omega\sim U(0,1)$, and calculate $$\alpha = \min\biggl[1,\frac{q^{\#}}{q^* } \times \frac{u_1(\theta^*|\theta^{\#},\Sigma_j)}{u_1(\theta^{\#}|\theta^{*},\Sigma_j)} 
\times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].$$ If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*=\theta^{\#}$ and $q^*:=q^{\#}$. If $\delta=\delta_{r_p}$ go to 5. Otherwise increase $j$ by 1 and if $j\in \{r_2,...,r_p\}$ update $\delta:=\delta_j$. Then go to 3. 5. Check current maximum (only when $\delta=\delta_{r_p}$): if $q^{\#}\pi(\theta^{\#})>\mathrm{max}\pi_{\delta_{r_p}}$ set $\mathrm{max}\pi_{\delta_{r_p}}:=q^{\#}\pi(\theta^{\#})$ and $\tilde{\theta}:=\theta^{\#}$. Increase $j$ by 1. If $j\leq s_1$ go to 3 else go to step 6. $5'.$ *(optional)* Apply regression adjustment on the draws obtained with $\delta_{r_p}$. Take as $\tilde{\theta}$ either the mean or the mode of the adjusted draws and call $\Sigma_{s_1}$ the covariance of the adjusted draws. **Data-cloning stage** 6. Take the last accepted value $\theta^*$, $\tilde{\theta}$ and the current covariance $\Sigma_{s_1}$ from ABC-MCMC. Set $\hat{\Sigma}_k:=\Sigma_{s_1}$, $K:=K_{s_2}$ and $\delta:=\delta_{r_p}$. 7. Generate $K$ independent vectors denoted $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$. Conditionally on each $X^{*(k)}$ generate a corresponding $z^{*(k)}$. Calculate $q^*=\prod_{k=1}^K J_{\delta}(y,z^{*(k)}).$ 8. Generate $\theta^{\#}:=\texttt{MIS}(\tilde{\theta},\hat{\Sigma}_k)$. Generate $K$ independent $X^{\#(k)}$’s from $p(X|\theta^{\#})$ and corresponding $z^{\#(k)}$. Compute $q^{\#}=\prod_{k=1}^K J_{\delta}(y,z^{\#(k)})$. 9. Generate $\omega\sim U(0,1)$ and calculate $$\alpha = \min\biggl[1,\frac{q^{\#}}{q^* }
\times  \frac{u_2(\theta^*|\tilde{\theta},\hat{\Sigma}_k)}{u_2(\theta^{\#}|\tilde{\theta},\hat{\Sigma}_k)} \times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].$$ If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*:=\theta^{\#}$ 