J_\delta(S(y),S(z^{(k)}))$.

Of course enlarging $K$ shrinks the area of the support of the cloned posterior where most of the probability mass is located, hence it becomes increasingly difficult to explore a progressively peaked surface: this is why in step (ii) of the schedule above we do not recommend to go below, say, 10-20% acceptance as this rate will reduce drastically when $K$ increases in step (iii).

In our applications we use a Gaussian kernel, that is $$J_{\delta}(y,z^{(k)}) \propto \exp\{-\bigl(S(z^{(k)})-S(y)\bigr)'\Omega^{-1}\bigl(S(z^{(k)})-S(y)\bigr)/2\delta^2\}
\label{eq:gauss-kernel}$$ where $'$ denotes transposition. Clearly equation respects desired criteria, i.e. (i) it is constant when $S(z^{(k)})\equiv S(y)$ and (ii) it gets larger values when $S(z^{(k)})\approx S(y)$. For a generic $z$ this implies writing $S(z)\sim N_{d_s}(S(y),\delta^2\Omega)$ with $N_{d_s}(\cdot,\cdot)$ a $d_s$-dimensional Gaussian distribution centred at $S(y)$ and $\Omega$ a positive definite matrix. For simplicity we assume a diagonal $\Omega$ with elements $\Omega=\mathrm{diag}\{\omega^{2}_1,...,\omega^{2}_{d_s}\}$. Of course, using a diagonal $\Omega$ might have an impact on the inference as it does not take into account the correlation among summary statistics. Recall that we keep writing $J_{\delta}(y,z)$ instead of $J_{\delta}(S(y),S(z))$, the dependence on data via summary statistics being considered as implicit.

When the elements in vector $S(\cdot)$ are varying approximately on the same range of values 