An identifiable model can have persistent ABC summary distortion

On either finite set, the discrete metric has values $0$ and $1$. 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. Each finite set is countable and dense in itself, and all singletons are open; thus its Borel sigma-algebra is the power set. The identity is a Borel isomorphism to this Polish presentation, so [F4] shows both spaces are standard Borel. By [F5], the usual metric on $\mathbb{R}$ is complete and separable and induces its usual Borel sigma-algebra, so the summary target is standard Borel. The constant map $S$ is measurable.

The sampling laws are $P_0={\delta }_0$ and $P_1={\delta }_1$, hence $P_0\mathrm{\ne }{P}_1$. By [F1], the identity parameter is identifiable. The marginal probability of the observation $y_0=0$ is $1\mathrm{/}2$; its prior-weighted likelihoods are $1\mathrm{/}2$ at parameter $0$ and $0$ at parameter $1$. Therefore the full-data conditional law is ${\delta }_0$, satisfying the posterior event identity [F7].

For every $ϵ\ge 0$, the event $\mathrm{\mid }S(Y)-0\mathrm{\mid }\le ϵ$ is the whole data space. It has probability one, so the ABC target is the unchanged prior, with mass $1\mathrm{/}2$ at each parameter. For rejection sampling supplied with the IID pairs of [F6], the first proposal is accepted and its output has this same law. The target's total-variation distance from the full-data posterior is at least the discrepancy on $\{0\}$, namely $\mathrm{\mid }1\mathrm{/}2-1\mathrm{\mid }=1\mathrm{/}2$, by [F7].

The statistic is not sufficient. If a common kernel $Q$ existed, taking summary event $C=\{0\}$ in [F2]'s identity would force $K(\theta ,D)=Q(0,D)$ for every data event $D$ and both parameters. For $D=\{0\}$ this would require both $Q(0,\{0\})=1$ and $Q(0,\{0\})=0$, a contradiction. Thus identifiability does not prevent persistent distortion from an insufficient summary.