MCMC algorithm execution, we let $K$ increase (see also [@doucet-godsill-robert]). A “dynamic” version of ABC-DC with varying $\delta$ and $K$ is presented in algorithm \[alg:abc-dc-dynam\]. Notice that previously cited work did not use ABC with data-cloning, so to the best of our knowledge ours is the first work proposing doing so.

As discussed by Christian P. Robert at <http://xianblog.wordpress.com/2010/09/22/feedback-on-data-cloning/> data-cloning share features with the simulated annealing global optimization method: keeping $K$ fixed to a high value once and for all removes the dynamic features of a simulated annealing random walk that first explores the whole space and then progressively focus on the highest modes, achieving convergence if the cooling is slow enough. In other words, if $K$ is “large enough”, the Metropolis algorithm will face difficulties in the exploration of the parameter space, and hence in the subsequent discovery of the global modes, while, if $K$ is “too small”, there is no certainty that the algorithm will identify the right mode of possibly multiple modes. Here follow a “dynamic” ABC-DC algorithm, where schedules are defined for $\delta$ and $K$, namely $\{\delta_{r_1},\delta_{r_2},...,\delta_{r_p}\}$ and $\{ K_{s_1},K_{s_2},...,K_{s_q}\}$ with $\delta_{r_1}>...>\delta_{r_p}>0$ and $1=K_{s_1}<...<K_{s_q}$. This version of ABC-DC starts with $s_1$ iterations of ABC-MCMC algorithm with 