Approximate Bayesian computation and the tolerance-conditioned target

Let $(\mathrm{\Theta },\mathcal{T})$ and $(\mathsf{Y},\mathcal{Y})$ be standard Borel spaces. Let $\mathrm{\Pi }$ be a prior probability measure on $(\mathrm{\Theta },\mathcal{T})$ and let $K$ be a probability kernel from $(\mathrm{\Theta },\mathcal{T})$ to $(\mathsf{Y},\mathcal{Y})$, as in Bayesian model as a prior and sampling kernel. Let $S:\mathsf{Y}\to {\mathbb{R}}^d$ be a measurable statistic for a fixed integer $d\ge 1$, and fix an observed summary $s_0\in {\mathbb{R}}^d$. The Euclidean norm is $\mathrm{\parallel }x{\mathrm{\parallel }}_2=({\sum }_{j=1}^d{x}_j^2{)}^{1\mathrm{/}2}$.

Under the prior-simulator joint law, first draw $\theta \sim \mathrm{\Pi }$ and then $Y\sim K(\theta ,\cdot )$. The summary-predictive probability measure $M$ on $\mathrm{\Theta }\times {\mathbb{R}}^d$ is the law of $(\theta ,S(Y))$; on measurable rectangles it is $M(A\times C)={\int }_{A}K(\theta ,S^{-1}(C))\mathrm{\Pi }(d\theta )\mathrm{.}$ The prior and simulator define a joint law by A prior and model kernel define a joint law, and the change-of-variables formula identifies $M$ as the law after applying $S$ (Change of variables for expectation).

For $ϵ\ge 0$, define the closed tolerance ball $B_{ϵ}(s_0)=\{s\in {\mathbb{R}}^d:\mathrm{\parallel }s-s_0{\mathrm{\parallel }}_2\le ϵ\}\mathrm{.}$ Its prior-predictive acceptance probability is $p_{ϵ}(s_0):=M(\mathrm{\Theta }\times B_{ϵ}(s_0))\in [0,1]\mathrm{.}$ Only when $p_{ϵ}(s_0)>0$ define the indicator ABC target by ${\mathrm{\Pi }}_{ϵ}(A\mid s_0):=\frac{M(A\times B_{ϵ}(s_0))}{p_{ϵ}(s_0)}(A\in \mathcal{T})\mathrm{.}$ This is exactly the conditional law of $\theta$ under the augmented event $S(Y)\in B_{ϵ}(s_0)$. The denominator is finite because $M$ is a probability, and no quotient is assigned when the event has probability zero. This target is the posterior given a tolerance event, not in general the full-data posterior or the posterior given the exact summary.

A positive tolerance separates summary information loss from tolerance bias; using finitely many accepted draws adds a separate Monte Carlo error. At a null summary value, a regular-conditional version may be changed there without affecting its defining identities, so its value is not determined by conditioning alone. The density regularity in ABC targets the density-defined summary posterior at zero tolerance selects the particular density-based target obtained as the positive-tolerance limit.