Revisiting concentration results for approximate Bayesian computation

Approximate Bayesian computation (ABC) has become a leading paradigm for drawing inference when data generating processes do not possess known or tractable likelihood functions. Recent expositions regarding the history, varieties, and example applications of ABC can be found in the comprehensive volume of Sisson et al. (2018). Other reviews of the ABC literature can be found in Marin et al. (2012) and Beaumont (2019). In this work, we study the asymptotic behaviors of two classes of ABC posterior measures, the so-called pseudo-posterior and coarsened posterior measures, in the context where the data generating process is unidentifiable. We now turn to describing those two classes of densities, the unidentifiable setting, as well as the possible asymptotic regimes in ABC.

Our work follows the progress of Jiang (2018), Miller and Dunson (2019), and Nguyen et al. (2020), who focus on the use of limit theorems to derive asymptotics regarding the ABC posteriors $\mathrm{\Pi }{}^{\mathrm{c}}{}_{n,ϵ}$ and $\mathrm{\Pi }{}^{\mathrm{p}}{}_{n,ϵ}$. This differs from the approaches of Frazier et al. (2018), Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2025), who rely on concentration of probability inequalities in order to provide rates at which the posterior measure converges in probability. However, although we do not provide in-probability rates, our results establish the stronger modes of $\text{L}{}_1$ and almost sure convergence of posterior objects instead.

As discussed in Section 5, it is also notable that Theorem 1 only provides posterior quasi-consistency results, rather than true posterior consistency, which would require a result that guarantees that $\mathrm{\Pi }{}^{\mathrm{c}}{}_{n,ϵ}$ or $\mathrm{\Pi }{}^{\mathrm{p}}{}_{n,ϵ}$ converges to a point mass, in some mode of convergence, when $ϵ$ is taken to be a decreasing function of $n$, as $n\to \mathrm{\infty }$. Such a result is in fact guaranteed by Frazier et al. (2018, Thm. 1), Bernton et al. (2019, Cor. 1), and Legramanti et al. (2025, Cor. 1), in probability. We note that Frazier et al. (2018, Thm. 1), Bernton et al. (2019, Cor. 1), and Legramanti et al. (2025, Cor. 1) make Hölder continuity assumptions that are stronger than our uniform continuity assumption when interpreting our assumption B2 in the identifiable case, and obviously much stronger than our assumptions in unidentifiable cases. In particular, in comparison to Frazier et al. (2018, Thm. 1), we do not require $D$ to be a distance between summary statistics. This is also true when comparing with the results of Li and Fearnhead (2018b, a).

Next, unlike Frazier et al. (2018), Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2025), we provide asymptotic results for weights $w$ other than the accept/reject kernel. This is a feature that is shared with Li and Fearnhead (2018b, a). However, our structural assumptions A3 and B4 are fairly mild in comparison to the location-scale form requirements and the existence of higher order moments of Li and Fearnhead (2018b, a). On the other hand, the additional restrictions together with stronger assumptions regarding the concentration of measure and weak convergence of summaries of $𝐗{}_n$ and $𝐘{}_n$, particularly the concentration of the prior measure $\mathrm{\Pi }$, permit Frazier et al. (2018); Li and Fearnhead (2018b, a), and Frazier et al. (2020) to establish weak and strong convergence of $\mathrm{\Pi }{}^{\mathrm{p}}{}_{n,ϵ}$ to measures with explicit forms.

Let $(\mathrm{\Omega },ℱ,\text{P})$ be a probability space with typical element $\omega$ and expectation operator $\mathrm{E}{}_{\mathrm{P}}$. We let $𝐗{}_{\mathrm{\infty }}=(X{}_i){}_{i\in \mathbb{N}}$ be a sequence of random variables $(X{}_i:\mathrm{\Omega }\to 𝕏\subset ℝ{}^d)$, and we denote the tuple containing the first $n$ elements of $𝐗{}_{\mathrm{\infty }}$ by $𝐗{}_n$, which is defined on the measurable space $(𝕏{}^n,ℬ(𝕏){}^{\otimes n})$. We define a measure on $𝐗{}_n$ via the density function $\text{p}{}_{\theta {}_0}(𝐱{}_n)$, dominated by the measure $𝔪{}^{\otimes n}$ (where $𝔪$ is typically the counting measure or Lebesgue measure), for some $\theta {}_0\in 𝕋$, which we call the generative parameter.

Next, we let $\theta$ denote a random parameter from the measure space $(𝕋,ℬ(𝕋),\mathrm{\Pi })$, where $\mathrm{\Pi }$ is characterized by its density $\pi$, dominated by the measure $𝔫$. Let $𝐘{}_{\mathrm{\infty }}=\left(Y{}_i\right){}_{i\in \mathbb{N}}$ ($Y{}_i:\mathrm{\Omega }\to 𝕏$) be a sequence of random variables with partial sequence $𝐘{}_n$ on $(𝕏{}^n,ℬ(𝕏){}^{\otimes n})$, and define the joint measure of $𝐘{}_n$ and $\theta$ via the density $\text{p}(𝐲{}_n|\theta )\pi (\theta )$, where $\text{p}(𝐲{}_n|\theta )$ is the likelihood of $𝐘{}_n$, and corresponds to a measure on $(𝕏{}^n,ℬ(𝕏){}^{\otimes n})$, for each $\theta \in 𝕋$. Throughout the text, we will take $𝐗{}_{\mathrm{\infty }}$ and $𝐘{}_{\mathrm{\infty }}$ to be independent, and take $𝕏$ and $𝕋$ to be Polish spaces. Furthermore, for fixed $\theta \in 𝕋$, we will write $\text{P}{}_{\theta }$ to denote the conditional measure of $\omega$ given $\theta$ that is compatible, in the sense of the Ionescu–Tulcea extension Theorem (Spataru, 2013, Thm. 16.8), with the measure on $𝐗{}_n$ and density $\text{p}(𝐲{}_n|\theta )$, for all $n\in \mathbb{N}$.

Define $w:ℝ{}_{\ge 0}\times ℝ{}_{>0}\to ℝ{}_{\ge 0}$, $(\delta ,ϵ)\mapsto w(\delta ,ϵ)$ to be a weighting or kernel function, where we typically think of $ϵ$ as a threshold parameter. For $𝐱{}_n,𝐲{}_n\in 𝕏{}^n$, we then let $(𝐱{}_n,𝐲{}_n)\mapsto D(𝐱{}_n,𝐲{}_n)\in ℝ{}_{\ge 0}$ denote some notion of a distance or a discrepancy between $𝐱{}_n$ and $𝐲{}_n$. We are now ready to describe our primary objects of interest, the pseudo-posterior density $\pi {}^{\mathrm{p}}{}_{ϵ,n}$

where

and the coarsened posterior density $\pi {}^{\mathrm{c}}{}_{ϵ,n}$

where

The pseudo-posterior density of form (1) is often studied in the context of importance sampling ABC (Karabatsos and Leisen, 2018; Nguyen et al., 2020) or kernel ABC (Park et al., 2016; Li and Fearnhead, 2018b, a). Of course, the accept/reject ABC algorithms (e.g. Marin et al. 2012, Algorithm 2) can be studied in this setting by taking $w(\delta ,ϵ)=\mathrm{𝟏}{}_{[0,ϵ)}\left(\delta \right)$, where we use the indicator function notation: $\mathrm{𝟏}{}_{𝔸}\left(x\right)=1$ if $x\in 𝔸$, and $\mathrm{𝟏}{}_{𝔸}\left(x\right)=0$ otherwise. Other popular choices of weights including the exponential kernel $w(\delta ,ϵ)=\exp (-|\delta |{}^{\alpha }/ϵ)$ (with $\alpha >0$) (Park et al., 2016), with Gaussian kernel special case, where $\alpha =2$, and the Epanechnikov kernel $w(\delta ,ϵ)=(3/4)\{1-(\delta /ϵ){}^2\}\mathrm{𝟏}{}_{[0,ϵ)}\left(\delta \right)$ (Beaumont et al., 2002). In Section 5, we also consider the tri-weight kernel $w(\delta ,ϵ)=\{1-(\delta /ϵ){}^2\}{}^3\mathrm{𝟏}{}_{[0,ϵ)}\left(\delta \right)$ and note that one may also validly choose any of the traditionally studied kernels from the theory of kernel density estimation, as per Scott (2015, Ch. 6). The purpose of the weight function is to generate a surrogate for the likelihood function that endows higher density on synthetic data that are closer to the observed data and lower density to those synthetic data that are further away. Here, the choice of weight function may be made on technical grounds, when properties such as continuity, differentiability, or support boundedness are often required, or to emulate other objects of interest, as per the measurement error interpretation of Wilkinson (2013).

Lastly, our study of the coarsened posterior density can be seen as a special case of that studied in Miller and Dunson (2019), under the assumption of correct specification of the simulation model, in the sense that there exists a $\theta {}_0\in 𝕋$ such that $𝐗{}_n$ has density $\text{p}(𝐱{}_n|\theta {}_0)$. We shall refer to both (1) and (2) as ABC posterior densities, when it is immaterial to distinguish between them.

Since we are operating on both the probability spaces $(\mathrm{\Omega },ℱ,\text{P})$ and $(𝕋,ℬ(𝕋),\mathrm{\Pi })$, it is important to maintain a clear nomenclature. Throughout the article, we will use the convention of denoting an event being true for almost every $\omega \in \mathrm{\Omega }$ by P-a.s. (P-almost sure/surely). Similarly, for an event that is true for almost all $\theta \in 𝕋$, we write $\mathrm{\Pi }$-a.a. ($\mathrm{\Pi }$-almost all/always).

Let the pseudo-posterior and coarsened posterior measures $\mathrm{\Pi }{}^{\mathrm{p}}{}_{ϵ,n}$ and $\mathrm{\Pi }{}^{\mathrm{c}}{}_{ϵ,n}$ be defined, respectively, as

for each $𝔸\in ℬ\left(𝕋\right)$ and $n\in \mathbb{N}$. We show in this section that $\mathrm{\Pi }{}^{\mathrm{p}}{}_{ϵ,n}$ and $\mathrm{\Pi }{}^{\mathrm{c}}{}_{ϵ,n}$ converge in total variation, almost surely and deterministically, respectively. To this end, we first establish the pointwise convergence of their densities in Proposition 1 below.

Let $D{}_{\mathrm{\infty }}(\cdot ,\cdot ):𝕋\times 𝕋\to ℝ{}_{\ge 0}$ and make the following assumptions:

A1

For $\mathrm{\Pi }$-a.a $\theta$, $D(𝐗{}_n,𝐘{}_n)\stackrel{[}{n}\to \mathrm{\infty }]\text{P}{}_{\theta }\text{-a.s.}⟶D{}_{\mathrm{\infty }}(\theta {}_0,\theta )$.

A2

For each $ϵ>0$, $sup{}_{\delta \in ℝ{}_{\ge 0}}w(\delta ,ϵ)<\mathrm{\infty }$.

A3

For each $ϵ>0$, $w\left(D{}_{\mathrm{\infty }}\right(\theta {}_0,\cdot ),ϵ):𝕋\to ℝ{}_{\ge 0}$ is continuous $\mathrm{\Pi }$-a.a.

Finally, we obtain the Q-a.s. convergence of the denominator of (1), by noticing that (3) implies that

by a second application of Hunt’s Lemma.  

Complete proofs of all results can be found in Appendix B.

Let us now discuss the assumptions. A1 holds in a variety of settings and is generally obtained by combining a strong law of large numbers and continuous mapping theorem. As applied in Section 5, a common situation that often appears (see, e.g., Marin et al., 2012) is when $D(𝐗{}_n,𝐘{}_n)=𝒟\left(\varsigma {}_n\right(𝐗{}_n),\varsigma {}_n(𝐘{}_n\left)\right)$, where, for each $n\in \mathbb{N}$, $\varsigma {}_n:𝕏{}^n\to 𝕊$ is a summary statistic that maps to the space $𝕊$, and $𝒟:𝕊\times 𝕊\to ℝ{}_{\ge 0}$ is continuous in both its arguments. Then, if the summary statistics are consistent in the sense that $\varsigma {}_n\left(𝐗{}_n\right)\stackrel{[}{n}\to \mathrm{\infty }]\text{P-a.s.}⟶\varsigma {}_{\theta {}_0}$ and $\varsigma {}_n\left(𝐘{}_n\right)\stackrel{[}{n}\to \mathrm{\infty }]\text{P}{}_{\theta }\text{-a.s.}⟶\varsigma {}_{\theta }$, for $\mathrm{\Pi }$-a.a $\theta$, then A1 holds with $D{}_{\mathrm{\infty }}(\theta {}_0,\theta )=𝒟(\varsigma {}_{\theta {}_0},\varsigma {}_{\theta })$. In Section 6 we also consider the case when $D(𝐗{}_n,𝐘{}_n)=\int {}_{ℝ}\left|\overline{F}{}_n\right(x;𝐗{}_n)-\overline{F}{}_n(x;𝐘{}_n\left)\right|\text{d}x$, where $\overline{F}{}_n(\cdot ;𝐗{}_n):ℝ\to [0,1]$ is the empirical distribution function corresponding to the sample $𝐗{}_n$ of observations supported on $𝕏=ℝ$. Then, by the Glivenko–Cantelli Theorem, it holds that $\overline{F}{}_n(\cdot ;𝐗{}_n)$ converges uniformly P-a.s to the distribution $F{}_{\theta {}_0}$ and similarly, $\overline{F}{}_n(\cdot ;𝐘{}_n)$ converges uniformly $\text{P}{}_{\theta }$-a.s to $F{}_{\theta }$. This then implies that $D(𝐗{}_n,𝐘{}_n)\stackrel{[}{n}\to \mathrm{\infty }]\text{P}{}_{\theta }\text{-a.s.}⟶D{}_{\mathrm{\infty }}(\theta {}_0,\theta )$, where $D{}_{\mathrm{\infty }}(\theta {}_0,\theta )=\int {}_{ℝ}\left|F{}_{\theta {}_0}\right(x)-F{}_{\theta }(x\left)\right|\text{d}x$, by Makarov and Podkorytov (2013, Thm. 4.8.1). Beyond these examples, we have that A1 holds, under appropriate conditions, when $D(𝐗{}_n,𝐘{}_n)$ is taken to be the energy distance (Nguyen et al., 2020), the sample Kullback–Leibler divergence (Jiang, 2018), the classification accuracy discriminant, or distances between posterior surrogates (Forbes et al., 2022).

Notice that no assumption is made regarding the independence between elements of the sequences $𝐗{}_{\mathrm{\infty }}$ and $𝐘{}_{\mathrm{\infty }}$. Further, no direct conditions are imposed on the limiting function $D{}_{\mathrm{\infty }}$, as it always appears composed with the weight function $w$, as in A3. The latter assumption requires continuity on a set of probability one with respect to the prior distribution, which is verified for the accept/reject kernel, for instance. Lastly, A2 is a simple requirement that the weight function $w$ is bounded.

We let $𝔹{}_{\theta }=\{\tau \in ℝ{}^q:\parallel \theta -\tau \parallel {}_2<1\}$ be the (open) unit ball in $ℝ{}^q$ with respect to the Euclidean norm $\parallel \cdot \parallel {}_2$, centered at $\theta \in ℝ{}^q$, and we note that we can scale $𝔹{}_0$ by a factor $\lambda >0$ to obtain balls $\lambda 𝔹{}_0=\{\lambda b:b\in 𝔹{}_0\}$ with any radius $\lambda$. Let $𝕋{}_0\subset 𝕋\subset ℝ{}^q$ be some set of interest. We say that $\mathrm{\Theta }{}_{\lambda }\subset 𝕋$ is a set of $\lambda$-covering centres for $𝕋{}_0$ if $𝕋{}_0\subset \mathrm{\Theta }{}_{\lambda }\oplus \lambda 𝔹{}_0$, where $𝔸\oplus 𝔹=\{a+b:a\in 𝔸,b\in 𝔹\}$ is the Minkowski sum. We will denote the cardinality of $\mathrm{\Theta }{}_{\lambda }$ by $\left|\mathrm{\Theta }{}_{\lambda }\right|$ and the Lebesgue measure on $ℝ{}^q$ by Leb.

Let $\theta {}_0\in 𝕋$. In the context of potential unidentifiability, consider the function $D{}_{\mathrm{\infty }}(\theta {}_0,\cdot ):𝕋\to ℝ{}_{\ge 0}$ and denote its set of zeros by $\mathrm{\Theta }{}_0=\{\theta \in 𝕋:D{}_{\mathrm{\infty }}(\theta {}_0,\theta )=0\}$. Then unidentifiability amounts to having a set of zeros $\mathrm{\Theta }{}_0$ of cardinality larger than 1. The illustrations of Section 6 will cover the identifiable case $|\mathrm{\Theta }{}_0|=1$, and then the unidentifiable cases $1<|\mathrm{\Theta }{}_0|<\mathrm{\infty }$ and $|\mathrm{\Theta }{}_0|=\mathrm{\infty }$.

We make the following assumptions:

B1

The set of zeros of $D{}_{\mathrm{\infty }}(\theta {}_0,\cdot ):𝕋\to ℝ{}_{\ge 0}$, denoted by $\mathrm{\Theta }{}_0$, is non-empty.

B2

For each $\lambda >0$, there exists an $ϵ>0$, such that $D{}_{\mathrm{\infty }}(\theta {}_0,\theta )<ϵ$ implies that $inf{}_{\tau \in \mathrm{\Theta }{}_0}\parallel \theta -\tau \parallel {}_2<\lambda$.

B3

For each $\lambda >0$, there exists a covering of $\mathrm{\Theta }{}_0$ with $\lambda$-covering centres $\mathrm{\Theta }{}_{\lambda }\subset 𝕋$, such that $\lambda {}^q\left|\mathrm{\Theta }{}_{\lambda }\right|\underset{\lambda \to 0}{⟶}0$.

B4

The weight function $w(\cdot ,\cdot )$ can be decomposed as $w(\delta ,ϵ)=W(\delta ,ϵ)\mathrm{𝟏}{}_{[0,ϵ)}\left(\phi \right(\delta \left)\right)$ where $W:ℝ{}_{\ge 0}\times ℝ{}_{\ge 0}\to ℝ{}_{\ge 0}$ and $\phi :ℝ{}_{\ge 0}\to ℝ{}_{\ge 0}$ is strictly increasing and bijective, with $\phi \left(0\right)=0$.

Let us now unpack the assumptions and conclusions of Theorem 1. Firstly, B1 simply assumes that the zeroes of $D{}_{\mathrm{\infty }}(\theta {}_0,\theta )=0$ exists. If $D{}_{\mathrm{\infty }}:𝕋\times 𝕋\to ℝ{}_{\ge 0}$ is a pseudometric on $𝕋$ (as per Richmond 2020, Sec. 11.1), then we may consider $\mathrm{\Theta }{}_0$ to be the equivalence class defined by the generative parameter: $\mathrm{\Theta }{}_0=\left[\theta {}_0\right]$, corresponding to the equivalence relationship: $D{}_{\mathrm{\infty }}(\theta {}_0,\theta )=0$.

B2 is a primitive boundedness and an identification assumption. A sufficient condition for B2 to hold is if the Hölder condition $inf{}_{\tau \in \mathrm{\Theta }{}_0}\parallel \theta -\tau \parallel {}_2\le LD{}_{\mathrm{\infty }}(\theta ,\theta {}_0){}^K$ is satisfied for each $\theta \in 𝕋$, for some $L,K>0$. This is further simplified when taking $\mathrm{\Theta }{}_0=\left\{\theta {}_0\right\}$ to be a singleton. Then, B2 is implied by condition that $\parallel \theta -\theta {}_0\parallel {}_2\le LD{}_{\mathrm{\infty }}(\theta ,\theta {}_0){}^K$, for every $\theta \in 𝕋$.

Next, B3 states that there exists a set of covering centres $\mathrm{\Theta }{}_{\lambda }$ of $\mathrm{\Theta }{}_0$ that does not grow too quickly, since we need the Lebesgue measure of the Euclidean balls covering $\mathrm{\Theta }{}_0$ to be small when $\lambda$ is small, in order to establish Part (i) of the theorem. Here, the assumption is automatically fulfilled when $\mathrm{\Theta }{}_0$ is a finite set, which particularly holds true when $\mathrm{\Theta }{}_0$ is a singleton. Note, however, that countability of $\mathrm{\Theta }{}_0$ is not a sufficient condition for B3 as in the case of dense countable subsets of $ℝ{}^q$. Taken together, B1–B3 require that there is a non-empty set of parameters that $\mathrm{\Theta }{}_0\subset 𝕋$ corresponding to the generative model of $𝐗{}_n$, that the sets of parameters in $𝕋$ that produce small discrepancies to the generative model are good approximations of the set $\mathrm{\Theta }{}_0$, and that the set $\mathrm{\Theta }{}_0$ can be covered by sets of arbitrarily small volume, respectively.

Lastly, B4 implies, when taken together with A2, that $w\left(D{}_{\mathrm{\infty }}\right(\theta {}_0,\cdot ),ϵ)$ and its support are bounded. This holds for the classical accept/reject kernel $w(\delta ,ϵ)=\mathrm{𝟏}{}_{[0,ϵ)}\left(\delta \right)$, but not for kernels with unbounded support, such as the Gaussian kernel $w(\delta ,ϵ)=\exp (-\delta {}^2/ϵ)$, that is considered in Park et al. (2016) and Nguyen et al. (2020). Note, however, that the conclusions of Proposition 1 remain true for kernels with unbounded supports.

We can interpret the results as follows. Lemma 1 states that under B1–B3, we can always cover the zero set $\mathrm{\Theta }{}_0$, of elements of $𝕋$ that are indistinguishable from $\theta {}_0$, using $D{}_{\mathrm{\infty }}$, by Euclidean balls with radius $\lambda$, such that the total volume of the covering vanishes with respect to the Lebesgue measure, as $\lambda \to 0$. Then, Theorem 1 states that if we further assume A1–A3 and B4, given any choice of $\lambda >0$, we can pick an $ϵ>0$ such that the pseudo-posterior and coarsened posterior measures, $\mathrm{\Pi }{}^{\mathrm{c}}{}_{ϵ,n}$ or $\mathrm{\Pi }{}^{\mathrm{p}}{}_{ϵ,n}$, of the covering of $\mathrm{\Theta }{}_0$ converge to full mass as $n\to \mathrm{\infty }$, almost surely and deterministically, respectively. That is, regardless of how small the Lebesgue measure of our covering of $\mathrm{\Theta }{}_0$ is, we can always choose an $ϵ>0$ such that the ABC posterior always eventually concentrates its mass entirely within the covering.

The conclusions can be viewed as a kind of posterior consistency, as defined in Ghosal and van der Vaart (2017, Ch. 6), where posterior consistency requires that the posterior measure concentrates on a point mass (with zero Lebesgue measure, in the continuous case), as $n\to \mathrm{\infty }$, in some sense, in the case where $\mathrm{\Theta }{}_0$ is a singleton. Here, we can call our conclusion a posterior quasi-consistency result, since we obtain that the ABC posterior measures concentrate on a set of negligible mass, instead, for potentially uncountable $\mathrm{\Theta }{}_0$.

In this section we illustrate examples covering different sets of zeros $\mathrm{\Theta }{}_0$. Firstly, in Section 6.1, $\mathrm{\Theta }{}_0$ is equal to the singleton $\{\theta {}_0\}$, and then in Section 6.2, $\mathrm{\Theta }{}_0$ is made of two points $\{\theta {}_0,-\theta {}_0\}$. Lastly, an example where $|\mathrm{\Theta }{}_0|=\mathrm{\infty }$ is studied in Section 6.3.

We have opted for examples with summary-based discrepancy functions $D$ and for IID sequences $𝐗{}_{\mathrm{\infty }}$ and $𝐘{}_{\mathrm{\infty }}$, for ease of understanding and simplicity of exposition. However, the main conclusions of Proposition 1 and Theorem 1 apply in much broader settings, as we will demonstrate below.