and we can safely interpret this step as the starting point for a new chain, targeting the corresponding powered distribution. The effect of step 7 is removed after some burnin period (and a presumably short one, as at this point the chain should already be in the bulk of the powered posterior). In the end, from the inference point of view, what matters are the draws generated at the largest $K$, which are generated with a fixed $\delta\equiv\delta_{r_p}$, hence are genuinely distributed according to the corresponding (marginal) posterior $\pi_\delta(\theta|y^{(K)})$.

For complex models a slow transition between different number of clones $K_{s_{j'}}$ might be required, meaning that differences $K_{s_{j'}}-K_{s_{j'+1}}$ should not be too large. This is because the chain needs to adapt to a narrower support for the posterior when using $K_{s_{j'+1}}$ clones, and to ease the exploration we use the covariance obtained from draws generated with $K_{s_{j'}}$ clones, which is hopefully appropriate. Otherwise if the difference above is too large the covariance used will be inappropriate (too large variances) and many proposals will be rejected.

A maximum likelihood ABC algorithm has also been proposed in [@rubio2013simple], however in that work a non-parametric kernel estimator of the posterior density is constructed 