Augmented ABC tolerance target

Use the prior and simulator setup of Approximate Bayesian computation and the tolerance-conditioned target: standard-Borel parameter and data spaces $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$, a prior probability $\mathrm{\Pi }$, a probability kernel $K(\theta ,\cdot )$, a measurable summary map $S:\mathsf{Y}\to {\mathbb{R}}^d$ with $d\ge 1$, an observed summary $s_0\in {\mathbb{R}}^d$, and tolerance $ϵ\ge 0$. Let $B_{ϵ}(s_0)=\{s\in {\mathbb{R}}^d:\mathrm{\parallel }s-s_0{\mathrm{\parallel }}_2\le ϵ\},A_{ϵ}=S^{-1}(B_{ϵ}(s_0))\mathrm{.}$ The prior-simulator joint law $H$ on the product sigma-algebra $(\mathrm{\Theta }\times \mathsf{Y},\mathcal{T}\otimes \mathcal{Y})$ is $H(C)={\int }_{\mathrm{\Theta }}K(\theta ,C_{\theta })\mathrm{\Pi }(d\theta ),C_{\theta }=\{y:(\theta ,y)\in C\},$ which is a probability measure by A prior and model kernel define a joint law. The finite product is standard Borel by Finite products of standard Borel spaces are standard Borel. Put $h_{ϵ}(\theta ,y)={\mathbf{1}}_{A_{ϵ}}(y),Z_{ϵ}={\int }_{\mathrm{\Theta }\times \mathsf{Y}}{h}_{ϵ}dH={\int }_{\mathrm{\Theta }}K(\theta ,A_{ϵ})\mathrm{\Pi }(d\theta )\mathrm{.}$ Thus $0\le Z_{ϵ}\le 1$, and $Z_{ϵ}$ is the prior-predictive probability $p_{ϵ}(s_0)$ in Approximate Bayesian computation and the tolerance-conditioned target. Only when $Z_{ϵ}>0$ define the augmented ABC tolerance target ${\tilde{\mathrm{\Pi }}}_{ϵ}(C)=\frac{1}{Z_{ϵ}}{\int }_C{h}_{ϵ}dH(C\in \mathcal{T}\otimes \mathcal{Y})\mathrm{.}$ It is the prior-simulator joint law conditioned on the summary-tolerance event. Its parameter marginal is identified in The augmented ABC target has the tolerance posterior as its parameter marginal. The definition does not assert that summary reduction or a positive tolerance preserves the full-data posterior.