and $q^*:=q^{\#}$. Increase $j$ by 1. If $j\in \{s_{3},...,s_{q}\}$ increase $K$ and update $\hat{\Sigma}_k:=\hat{cov}(\theta)_{k'}$ then go to 7, otherwise go to 10. Here $\hat{cov}(\theta)_{k'}$ is the sample covariance computed on draws generated using the previous value $k'$ of $K$ in the schedule. 10. If $j\leq R$ go to 8 otherwise stop.

\[alg:abc-dc-dynam\]

During ABC-MCMC we generate parameter proposals using adaptive Gaussian Metropolis random walk [@haario-saksman-tamminen]. We write $\theta^{\#}:= \texttt{AMRW}(\theta^*,\Sigma_j)$ to denote such a proposal (and the corresponding Gaussian proposal function is denoted $u_1(\cdot)$). During the exploration of the approximate posterior surface with $K=1$ we aim at locating the principal mode $\tilde{\theta}$ of the distribution for the smallest threshold $\delta_{r_p}$ as described above. At the end of the $s_1$ iterations of ABC-MCMC we fix $\delta:=\delta_{r_p}$ for the rest of the ABC-DC execution and increase $K$ progressively. At this stage parameters are proposed using a Metropolis independent sampler $\texttt{MIS}(\tilde{\theta},\hat{\Sigma}_k)$ generating from the Gaussian distribution $N(\tilde{\theta},\hat{\Sigma}_k)$ (and the corresponding Gaussian proposal function is denoted $u_2(\cdot)$) where $\hat{\Sigma}_k$ is the sample covariance matrix obtained from draws generated using the previous value of $K$.

We switched from a Gaussian random walk to an independent Metropolis sampler when powering-up because ABC-MCMC should have located the bulk 