that ABC methods are most often applied to models where a set of low-dimensional summaries of the data $S(y)$ is employed rather than the full dataset $y$. That is whenever is possible (and even more so when $S(\cdot)$ is sufficient for $\theta$), it is advisable to consider $J_{\delta}(S(y),S(z))$ instead, so that for example we have $$\pi_{\delta}(\theta,z|\rho(S(z),S(y))\leq\delta)\propto L(\theta;z)\pi(\theta)\mathbb{I}_{\{\rho(S(z),S(y))\leq\delta\}}$$ when $J_{\delta}(S(y),S(z))\propto \mathbb{I}_{\{\rho(S(z),S(y))\leq\delta\}}$.

The introduction of summaries $S(\cdot)$ is a double edged sword. On one hand the specification of appropriate (i.e. informative, though usually not sufficient) statistics is not trivial, especially for dynamic models, whereas it is somehow more intuitive to specify them for static models. On the other hand having an informative set of statistics implies a considerable reduction in the number of elements to be compared ($d_s$ comparisons, where $d_s:=\dim(S)$ instead of the $n$ comparisons required when simulated and observed data have to be compared, with $d_s\ll n$) and consequently a much smaller $\delta$ can be employed.

ABC-DC: Data-cloning ABC {#sec:abc-dc}
------------------------

In most problems $\delta$ is a strictly positive value, sometimes set “small enough”, some other times set to a larger than desired value, depending on the complexity of the experimental scenario. In fact when using a very small $\delta$ 