Vanishing acceptance probability with a stable zero-tolerance target

Give $\mathrm{\Theta }$ its discrete metric. Since Cauchy means that tail distances are eventually below every positive radius, a Cauchy sequence is eventually constant at radius $1\mathrm{/}2$, so the metric is complete; the finite set is countable and dense in itself, and its singletons are open. Its Borel sigma-algebra is the power set, so [F3] gives standard Borelness. By [F2], the usual metric on $\mathbb{R}$ is complete and has countable dense subset $\mathbb{Q}$; its balls are ordinary intervals, so it induces the usual topology. Thus both the data and summary Borel spaces are standard Borel by [F3]. Each displayed density is nonnegative measurable and integrates to one because its supporting interval has length one. Thus $K(\theta ,ds)=m_{\theta }(s)ds$ is a probability kernel on the finite parameter space, and the identity summary is measurable. For every product-measurable $E\subseteq \mathrm{\Theta }\times \mathbb{R}$, the section maps $s\mapsto (\theta ,s)$ are measurable, so $E_{\theta }=\{s:(\theta ,s)\in E\}$ is measurable, and $E={⨆}_{\theta \in \mathrm{\Theta }}\{\theta \}\times E_{\theta }$. Since $\mathrm{\Theta }$ is finite, the rectangle identity extends by this finite disjoint decomposition to $M(E)={\sum }_{\theta \in \mathrm{\Theta }}\mathrm{\Pi }(\{\theta \}){\int }_{E_{\theta }}{m}_{\theta }(s)ds={\int }_E{m}_{\theta }(s)(\mathrm{\Pi }\times {\lambda }_1)(d\theta ,ds)\mathrm{.}$ Thus the finite-parameter summary-predictive law has the specified joint density required by the zero-tolerance theorem.

For $0<ϵ<1\mathrm{/}2$, the tolerance ball about $1\mathrm{/}2$ is $[1\mathrm{/}2-ϵ,1\mathrm{/}2+ϵ]$. Its intersection with $[0,1]$ has length $2ϵ$, while it is disjoint from $[1,2]$ because its upper endpoint is strictly below $1$. The parameter-conditional acceptance probabilities are therefore $2ϵ$ at $0$ and $0$ at $1$. Averaging against the equal prior gives $p_{ϵ}=ϵ>0$, and conditioning gives ABC target ${\delta }_0$. In particular $p_{ϵ}\to 0$. By [F5], the accepted parameter also has law ${\delta }_0$.

On the neighborhood $[1\mathrm{/}4,3\mathrm{/}4]$ of $s_0=1\mathrm{/}2$, $m_0$ is identically $1$ and $m_1$ is identically $0$. Both densities are continuous at $s_0$, and the constant envelope $H(\theta )=1$ is integrable under $\mathrm{\Pi }$. The averaged density is $z_0=\frac{1}{2}{m}_0(1\mathrm{/}2)+\frac{1}{2}{m}_1(1\mathrm{/}2)=1\mathrm{/}2$, so the density-selected posterior is ${\delta }_0$.

The hypotheses of [F6] hold under the Countable Choice assumption [F4] (with, for example, neighborhood radius $1\mathrm{/}4$), so the ABC targets converge in total variation to ${\delta }_0$ as $ϵ\downarrow 0$. Here step 2.1 shows the stronger exact identity ${\mathrm{\Pi }}_{ϵ}={\delta }_0$ for every $0<ϵ<1\mathrm{/}2$, even though the acceptance probability vanishes.