parameters are integrated out.

The paper is organized as follows: in section \[sec:data-cloning\] we briefly review some properties of “data cloning” for maximum likelihood estimation; in section \[sec:ABC-all\] we introduce topics of approximate Bayesian computation methodology, first by considering some basics (section \[sec:abc-basics\]) then introducing our original contribution in sections \[sec:abc-dc\]–\[sec:dynamic-abcdc\]. Finally in section \[sec:simulations\] simulation studies illustrate results.

Data cloning {#sec:data-cloning}
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A Bayesian procedure for maximum likelihood estimation based on “replicating” data was first introduced in [@robert1993] (see also [@robert-titterington]) and then studied under different flavors by e.g. [@doucet-godsill-robert], [@jacquier-johannes-polson] and [@lele-dennis-lutscher]. The latter introduced the term “data-cloning” which we employ. For simplicity, in the following we always consider the full vector parameter $\theta$, and it should be understood that some of its components enter $f(\cdot)$ while others enter $g(\cdot)$. The likelihood function of $\theta$ for the state-space model can be written as $$\begin{aligned}
L(\theta;y)&=p(y_0;\theta)\prod_{i=1}^np(y_i|y_1,...,y_{i-1};\theta)\nonumber\\
&=
\int f(y_0|x_0;\theta)p(x_0) \prod_{i=1}^n \bigl\{ f(y_i|x_i,\theta)g(x_i|x_{i-1},\theta)\bigr\}dx_0\cdots dx_n
\label{eq:likelihood}\end{aligned}$$ where $x_0=X_{t_0}$ and $p(x_0)$ the corresponding unconditional density. Actually in what follows, and without loss of generality, we assume $x_0$ deterministic and known, hence we remove $p(x_0)$ from the expression of the likelihood function. Notice the latter equality in exploits the notion of conditional independence 