to obtain accurate inference, this results in a high rejection rate, often too high to be computationally feasible. Therefore we propose to consider a larger $\delta$ than what would typically be considered as appropriate, coupled with data-cloning. Our idea is that if we use $K=1$ while dynamically decrease $\delta$ in an ABC-MCMC algorithm, to reach a moderately large $\delta$-value that still allows exploration of the posterior surface (producing an acceptance rate of, say, 10-15%, and this phase could be considered as “burn-in”) we can then start a data-cloning procedure and progressively enlarge $K$ while keeping $\delta$ constant to its last value. During the initial exploration (burn-in with $K=1$) we require a $\delta$ which is small enough to locate the approximate position of the maxima for the exact marginal posterior $\pi(\theta|y)$, not a $\delta$ producing an accurate approximation to the surface of $\pi(\theta|y)$. When we start increasing $K$ the marginal posterior $\pi_{\delta}(\theta|y^{(K)})$ will concentrate around its maxima, which for large enough $K$ should be approximately at the same location as the MLE.

More in detail we propose to consider $R$ iterations of an ABC-MCMC algorithm later denoted as ABC-DC: (i) start an ABC-MCMC algorithm without data-cloning ($K=1$) and let $\delta$ decrease 