the “static” ABC-DC is given in algorithm \[alg:abc-dc\], where both $\delta$ and $K$ are assumed fixed, allowing for a more immediate understanding. However, in our applications we use algorithm \[alg:abc-dc-dynam\] which is discussed later.

1\. Initialization: Fix a starting value $\theta^*$ or generate it from its prior $\pi(\theta)$ and set $\theta_1=\theta^*$. Set $j=1$, fix $\delta>0$ and a positive integer $K$. A vector of statistics $S(\cdot)$ and weights $\Omega$ is available. 2. Generate $K$ independent values of the latent process $X^{*(1)},...,X^{*(K)}$ from $p(X|\theta^{*})$. Conditionally on each $X^{*(k)}$ generate a corresponding $z^{*(k)}$ from equation for each $k=1,...,K$. 3. Calculate $J_{\delta}(y,z^{*(k)})$ for every $k$ and compute $$q^*=\prod_{k=1}^K J_{\delta}(y,z^{*(k)}).$$ 4. Generate a $\theta^{\#}\sim u(\theta^{\#}|\theta^*)$. Generate $K$ independent $X^{\#(k)}$’s from $p(X|\theta^{\#})$ and corresponding $z^{\#(k)}$. 5. Calculate $J_{\delta}(y,z^{\#(k)})$ for every $k$ and set $q^{\#}=\prod_{k=1}^K J_{\delta}(y,z^{\#(k)})$. Generate $\omega\sim U(0,1)$, and calculate $$\alpha = \min\biggl[1,\frac{q^{\#}}{q^* }
\times  \frac{u(\theta^*|\theta^{\#})}{u(\theta^{\#}|\theta^{*})} \times \frac{\pi(\theta^{\#})}{\pi(\theta^*)} \biggr].$$ If $\omega>\alpha$, set $\theta_{j+1}:=\theta_{j}$ and, increase $j$ by 1 and go to step 6. Otherwise set $\theta_{j+1}:=\theta^{\#}$, $\theta^*:=\theta^{\#}$, $q^*:=q^{\#}$, increase $j$ by 1 and go to 6. 6. Repeat steps 4–5 as long as $j\leq R$.

\[alg:abc-dc\]

Dynamic ABC-DC {#sec:dynamic-abcdc}
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In our experiments, and unlike in [@lele-dennis-lutscher] and [@jacquier-johannes-polson] where $K$ is kept fixed during the 