during this phase; an initially large $\delta$ will enable a rapid exploration of the posterior surface at a high acceptance rate (say 30%) to locate the bulk of the approximated posterior. (ii) while $\delta$ progressively decreases, the algorithm focus on exploring a more accurate approximation of the posterior until $\delta$ reaches say 10-20% acceptance rate. Such acceptance rate is typically too high in ABC studies (a typical value would be 1% or less) however we plan to increase the number of clones and focus on the peak of such posterior. (iii) At this point $\delta$ is kept fixed to its last value and data-cloning starts, by progressively increasing the value of $K$. (iv) Once $R$ iterations are completed, we collect the draws generated with the largest value of $K$ and use those for maximum likelihood inference. As such the resulting samples are not from $\pi_{\delta}(\theta,z|y)$ but from the powered-approximated posterior $\pi_{\delta}(\theta,z^{(1)},...,z^{(K)}|y^{(K)})$ for finite $K$ $$\begin{aligned}
\pi_{\delta}(\theta,z^{(1)},...,z^{(K)}|y^{(K)}) \propto \pi(\theta)\prod_{k=1}^K J_\delta(y,z^{(k)})L(\theta;z^{(k)})\end{aligned}$$ where the $L(\theta;z^{(k)})$ can be simplified out in the Metropolis-Hastings acceptance probability as previously illustrated. As mentioned in section \[sec:abc-basics\], we assume as implicit the dependence on data via summary statistics, therefore here and in the rest of our work $J_\delta(y,z^{(k)})\equiv 