be postulated, as described below. In ABC we aim at simulating draws from the augmented approximated posterior $$\pi_{\delta}(\theta,z|y)\propto J_{\delta}(y,z)\underbrace{L(\theta;z)\pi(\theta)}_{\propto \pi(\theta|z)}
\label{eq:abc-posterior}$$ where $z=(z_0,...,z_n)$ and $L(\theta;z)$ is the (intractable) likelihood function for $\theta$ based on $z$. Then $\pi_{\delta}(\theta|y)\propto \int \pi_{\delta}(\theta,z|y)dz$. Here $J_{\delta}(\cdot)$ is some function that depends on $\delta$ and weights the intractable posterior for simulated data $\pi(\theta|z)\propto L(\theta;z)\pi(\theta)$ with high values in regions where $z$ and $y$ are similar. Therefore we would like (i) $J_{\delta}(\cdot)$ to give higher rewards to proposals corresponding to $z$ having values close to $y$. In addition (ii) $J_{\delta}(y,z)$ is assumed to be a constant when $z=y$ (i.e. when $\delta=0$) so that the *exact* marginal posterior $\pi(\theta|y)$ is recovered. A common choice for $J_{\delta}(y,z)$ is the uniform kernel $$J_{\delta}(y,z)\propto \mathbb{I}_{\{\rho(z,y)\leq\delta\}}$$ where $\rho(z,y)$ is some measure of closeness between $y$ and $z$ and $\mathbb{I}$ is the indicator function. Important alternatives are the Epanechnikov and Gaussian kernels [@beaumont2010approximate] or sums of discrepancies [@toni2009approximate]. An ABC-MCMC algorithm targeting the distribution has been proposed in [@marjoram2003markov]. However, one of the difficulties is that, in practice, $\delta$ has to be set as a tradeoff between statistical accuracy (with a small positive $\delta$) and computational feasibility ($\delta$ not too small). Also, notice 