mechanism. Depending on the modelling scenario, $Y$ might be observed conditionally on an unobservable random variable $X$, i.e. $Y\sim f(y|X,\phi)$, or $X$ might be fixed and known (e.g. a set of deterministic inputs or covariates). In any case, there is dependence on an unknown (vector) parameter $\phi$. The statistical methods we are going to introduce have general appeal, however in order to set a working framework for the time being we assume to deal with state-space models (also known as Hidden Markov Models, [@cappe2005inference]). We remark that our methods based on ABC are *not* restricted to state-space, nor dynamic models. For example, below we assume $X$ a Markov process, but this is not a requirement nor is conditional independence of measurements.

Consider an observable, discrete-time stochastic process $\{Y_t\}_{t\geq t_0}$, $Y_t\in\mathsf{Y}\subseteq \mathbb{R}^{d_y}$ and a latent and unobserved continuous-time stochastic process $\{X_t\}_{t\geq t_0}$, $X_t\in\mathsf{X}\subseteq \mathbb{R}^{d_x}$. Process $X_t\sim g(x_t|x_{t-1},\eta)$ is assumed Markov with transition densities $g(\cdot)$ depending on another unknown (vector) parameter $\eta$. We think at $\{Y_t\}$ as a measurement-error-corrupted version of $\{X_t\}$ and assume that observations for $\{Y_t\}$ are conditionally independent given $\{X_t\}$. Our state-space model can be summarised as $$\begin{cases}
Y_t\sim f(y_t|X_t,\phi),\qquad t\geq t_0\\
X_t\sim g(x_t|x_{t-1},\eta).
\end{cases}
\label{eq:state-space-general}$$ Typically $f(\cdot)$ is 