ABC rejection sampling

The inputs are a proper prior $\mathrm{\Pi }$ on $(\mathrm{\Theta },\mathcal{T})$, a simulator probability kernel $K(\theta ,dy)$, a measurable summary statistic $S:\mathsf{Y}\to {\mathbb{R}}^d$, an observed summary $s_0$, and a tolerance $ϵ\ge 0$ as in Approximate Bayesian computation and the tolerance-conditioned target. The procedure is admissible when $p_{ϵ}(s_0)>0$.

1. Draw a candidate parameter $\theta$ from $\mathrm{\Pi }$.
2. Conditional on $\theta$, draw synthetic data $Y$ from $K(\theta ,\cdot )$, independently of all earlier proposal-simulator pairs.
3. Compute $S(Y)$. If $\mathrm{\parallel }S(Y)-s_0{\mathrm{\parallel }}_2\le ϵ$, return $\theta$ and stop. Otherwise repeat from step 1 with a fresh independent pair.

The output is one accepted parameter value. It does not itself assert a finite-run estimate or an accuracy bound for an empirical distribution of accepted values.