# Concentration results for approximate Bayesian computation without identifiability

Hien Nguyen, Trungtin Nguyen, Julyan Arbel, Florence Forbes

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Hien Nguyen, Trungtin Nguyen, Julyan Arbel, Florence Forbes. Concentration results for approximate Bayesian computation without identifiability. 2023. hal-03987197

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# Revisiting Concentration Results for Approximate Bayesian Computation

Hien Nguyen, Trungtin Nguyen, Julyan Arbel, Florence Forbes

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Hien Nguyen, Trungtin Nguyen, Julyan Arbel, Florence Forbes. Revisiting Concentration Results for Approximate Bayesian Computation. Bayesian Analysis, 2025, pp.1-23. 10.1214/25-BA1520. hal-03987197v2

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Distributed under a Creative Commons Attribution 4.0 International License
# Concentration results for approximate Bayesian computation
without identifiability

BY H. D. NGUYEN

*School of Mathematics and Physics, Univ. of Queensland*
*St. Lucia 4067, Queensland, Australia.*

h.nguyen7@uq.edu.au

T. T. NGUYEN, J. ARBEL, AND F. FORBES

*Univ. Grenoble Alpes, Inria, CNRS*
*Grenoble INP, LJK 38000 Grenoble, France.*

trung-tin.nguyen@inria.fr julyan.arbel@inria.fr florence.forbes@inria.fr

## SUMMARY

We study the large sample behaviors of approximate Bayesian computation (ABC) posterior measures in situations when the data generating process is dependent on non-identifiable parameters. In particular, we establish the concentration of posterior measures on sets of arbitrarily measure that contain the equivalence set of the data generative parameter, when the sample size tends to infinity. Our theory also makes weak assumptions regarding the measurement of discrepancy between the data set and simulations, and in particular, does not require the use of summary statistics and is applicable to a broad class of kernelized ABC algorithms. We provide useful illustrations and demonstrations of our theory in practice, and offer a comprehensive assessment of the nature in which our findings complement other results in the literature.

*Some key words:* Approximate Bayesian computation; Posterior consistency; Pseudo-posterior measure; Non-identifiable; Concentration of mass; Large sample theory

## 1. INTRODUCTION

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).

We shall place our investigation within the following general setting. Let $(\Omega, \mathcal{F}, P)$ be a probability space with element $\omega$ and expectation operator $E$. We let $X_{\infty} = (X_i)_{i \in \mathbb{N}}$ be a sequence of random variables, where $X_i : \Omega \to \mathcal{X} \subset \mathbb{R}^d$ and $X_n$ is a tuple containing the first $n$ elements of $X_{\infty}$ and has probability measure on the space $(\mathcal{X}^n, \mathcal{B}(\mathcal{X}^n))$ which is absolutely continuous with respect to $\mathfrak{m}$ (typically the counting or Lebesgue measure), defined by density $p(x_n|\theta_0)$, for some $\theta_0 \in \mathbb{T} \subset \mathbb{R}^q$ $(i, d, n, q \in \mathbb{N})$, which we shall call the *generative parameter*. We further let $\mathbf{Y}_{\infty} = (Y_i)_{i \in \mathbb{N}}$ be another sequence of random variables $Y_i : \Omega \to \mathcal{X}$, where the partial sequence $\mathbf{Y}_n$ also has a probability measure on $(\mathcal{X}^n, \mathcal{B}(\mathcal{X}^n))$ with density $p(\mathbf{y}_n|\theta)$ and $\theta \in \mathbb{T}$. Throughout, $X_{\infty}$ and $\mathbf{Y}_{\infty}$ are taken to be independent.
# Revisiting Concentration Results for Approximate Bayesian Computation

Hien Duy Nguyen\*,†, TrungTin Nguyen‡, Julyan Arbel§, and Florence Forbes§

**Abstract.** We study the large sample behavior of approximate Bayesian computation (ABC) posterior measures in situations when the data generating process is dependent on unidentifiable parameters. In particular, we establish the concentration of posterior measures on sets of arbitrarily small measure that contain the equivalence set of the data generative parameter, when the sample size tends to infinity. Our theory also makes weak assumptions regarding the measurement of discrepancy between the data set and simulations. In particular, it does not require the use of summary statistics and is applicable to a broad class of kernelized ABC algorithms. We provide useful illustrations and demonstrations of our theory in practice, and offer a comprehensive assessment of how our findings complement other results in the literature.

**Keywords:** approximate Bayesian computation, posterior consistency, pseudo-posterior measure, unidentifiability, large sample theory.

**MSC2020 subject classifications:** Primary 62E20, 62F15; secondary 60B10.

## 1 Introduction

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.

**Pseudo-posterior density** The density obtained by a usual ABC algorithm is referred to here as the *pseudo-posterior density*. A typical ABC sampling scheme consists of three steps:

1. sample some parameter value $\theta$ from the prior distribution;

2. sample some synthetic data from the data generative process, given $\theta$;

<footnote>\*School of Computing, Engineering and Mathematical Sciences, La Trobe Univ., Bundoora 3086, Victoria Australia, h.nguyen5@latrobe.edu.au</footnote>

<footnote>†Institute of Mathematics for Industry, Kyushu Univ., Nishi Ward, Fukuoka 819-0395, Japan</footnote>

<footnote>‡School of Mathematical Sciences, Queensland Univ. of Technology, Brisbane City 4000, Queensland Australia</footnote>

<footnote>§Univ. Grenoble Alpes, Inria, CNRS, Grenoble INP, LJK 38000 Grenoble, France</footnote>
Let $w(\cdot, \cdot) : \mathbb{R}_{\ge 0} \times \mathbb{R}_{> 0} \to \mathbb{R}_{\ge 0}$, $(\delta, \epsilon) \mapsto w(\delta, \epsilon)$ be a weighting or kernel function, where we typically think of $\epsilon$ as a threshold parameter. For $\mathbf{x}_n, \mathbf{y}_n \in \mathbb{X}^n$, we then let $(\mathbf{x}_n, \mathbf{y}_n) \mapsto D(\mathbf{x}_n, \mathbf{y}_n) \in \mathbb{R}_{\ge 0}$ denote some notion of a distance or a discrepancy between $\mathbf{x}_n$ and $\mathbf{y}_n$ and further let $\theta$ be a random element on the probability space $(\mathbb{T}, \mathcal{B}(\mathbb{T}), \Pi)$, where $\Pi$ is absolutely continuous with respect to measure $n$, with density $\pi(\theta)$. We concern ourselves with studying the behavior of the pseudo-posterior density:

$$ \pi_{\epsilon}(\theta|\mathbf{X}_n) = \frac{\pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)}{\mathcal{C}_{\epsilon}(\mathbf{X}_n)}, \epsilon \in \mathbb{R}_{>0}, \quad (1) $$

where

$$ \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) = \int_{\mathbb{X}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) p(\mathbf{y}_n|\theta) \mathrm{d}\mathbf{m}(\mathbf{y}_n) $$

and

$$ \mathcal{C}_{\epsilon}(\mathbf{X}_n) = \int_{\mathbb{T}} \pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \mathrm{d}\mathbf{n}(\theta). $$

We shall also consider the coarsened-posterior density:

$$ \bar{\pi}_{\epsilon,n}(\theta) = \frac{\pi(\theta) \mathrm{E}\{\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)\}}{\bar{\mathcal{C}}_{\epsilon,n}}, \epsilon \in \mathbb{R}_{>0}, \quad (2) $$

where

$$ \bar{\mathcal{C}}_{\epsilon,n} = \int_{\mathbb{T}} \pi(\theta) \mathrm{E}\{\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)\} \mathrm{d}\mathbf{n}(\theta). $$

We note that the pseudo-posterior density of form (1) is often studied in the context of importance sampling ABC (Karabatsos & Leisen, 2018; Nguyen et al., 2020) or kernel ABC (Park et al., 2016; Li & Fearnhead, 2018b,a) algorithms. Our study of the coarsened-posterior density (2) can be viewed as a homage to and generalization of results from Miller & Dunson (2019), who consider the case where $\mathbf{X}_{\infty} = \mathbf{x}_{\infty}$, with probability 1, for some fixed sequence $\mathbf{x}_{\infty}$ (cf. Miller & Dunson, 2019, Eqn. 2.1). Of course, the usual accept/reject ABC algorithms (e.g. Marin et al. 2012, Algorithm 2) can be studied in this setting by taking $w(\delta, \epsilon) = [[\delta < \epsilon]]$, where we use the Iverson bracket notation: $[[A]] = 1$ if statement A is true, and $[[A]] = 0$ otherwise. We shall refer to both (1) and (2) as ABC posterior densities, when it is immaterial to distinguish between them.

In recent works the asymptotic behaviors of (1) has been investigated by Bernton et al. (2019), Nguyen et al. (2020), and Legramanti et al. (2022), when $\epsilon \to 0$, for fixed $n$; Jiang (2018), and Nguyen et al. (2020), when $\epsilon > 0$ is fixed and $n \to \infty$; and Frazier et al. (2018), Li & Fearnhead (2018b), Li & Fearnhead (2018a), Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2022), when $\epsilon \to 0$ and $n \to \infty$. Miller & Dunson (2019) provide analysis for the behavior of (2) when $\epsilon > 0$ is fixed and $n \to \infty$, in the special case when $\mathbf{X}_{\infty} = \mathbf{x}_{\infty}$. Our current work extends upon the best known results when $\epsilon > 0$ and $n \to \infty$. Namely, we prove broadly applicable results regarding the concentration of mass of the pseudo-posterior and coarsened-posterior measures $\Pi_{\epsilon,n}$ and $\bar{\Pi}_{\epsilon,n}$ (collectively, ABC posterior measures) as $n \to \infty$, in practical cases when the densities $p(\mathbf{x}_n|\theta)$ are not identifiable with respect to $\theta \in \mathbb{T}$. We shall elaborate in the sequel.
3. compute some distance between the synthetic data and the actual data in order to treat $\theta$.

Multiple ABC variants exist depending on the treatment of the parameter in Step 3. The usual accept/reject ABC algorithm (e.g. Marin et al. 2012, Algorithm 2) discards parameter samples that are at a distance beyond some threshold $\epsilon$, while keeping all the parameters below the threshold. In contrast, variants known as importance sampling ABC (Karabatsos and Leisen, 2018; Nguyen et al., 2020) or kernel ABC (Park et al., 2016; Li and Fearnhead, 2018b,a) yield weighted ABC posterior samples, where sample weights are computed by taking kernel function transformations of sample discrepancy values. Refer to Equation (1) in Section 3 for a detailed description of the pseudo-posterior density, and more specifically to the description of weights at the end of Section 3.

**Coarsened posterior density** The approach followed by Miller and Dunson (2019) is different in nature from ABC, however we will see in Section 3 that it can be described in similar statistical terms. The motivation of these authors stems from realizing that the distribution of the observed data rarely belongs to the chosen model class of idealized data. In order to ensure robustness to model misspecification, the basic idea proposed by Miller and Dunson (2019) in a Bayesian context is to condition on the event that the observed data and idealized data are sufficiently close, resulting in the so-called *coarsened posterior density*. The degree of closeness is defined according to the distance between the empirical measures of the observed data and idealized data, assumed to be below some threshold denoted by $r$ in Miller and Dunson (2019), and which will be denoted by $\epsilon$ here. Refer to Equation (2) in Section 3 for a detailed description of the coarsened posterior density.

**Unidentifiability** We are interested in models which are not necessarily identifiable, meaning that multiple parameter values may result in the same likelihood. Let us denote by $\theta_0$ the true data generating parameter. The meaning of such a true parameter in a Bayesian context may seem counterintuitive, but such a value is often postulated in works that endeavour to validate posterior distributions based on frequentist principles in the asymptotic regime of large sample size. Let $\Theta_0$ be the set of parameters which identify the same likelihood as the true data generating parameter $\theta_0$. Then the model is identifiable if $\Theta_0$ reduces to the singleton $\{\theta_0\}$, and unidentifiable otherwise. Illustrations of identifiable, as well as finitely unidentifiable, and infinitely unidentifiable normal models are presented in Sections 6.1, 6.2, and 6.3, respectively.

**Asymptotic settings for ABC posterior measures** The asymptotic behavior of the ABC posterior can be investigated in three scenarios depending on the regime of the tolerance parameter $\epsilon$ and the sample size $n$.

① $\epsilon$ fixed and $n \to \infty$: when the tolerance parameter is held fixed while the sample size increases to infinity, the ABC posterior measures are expected to converge to some pseudo-posterior distribution. This scenario is particularly relevant when computational resources allow for larger sample sizes and large data sets are available.
## 2. WEAK CONVERGENCE OF ABC POSTERIOR MEASURES

Since we are operating on both the measure spaces $(\Omega, \mathcal{F}, \mathbb{P})$ and $(\mathbb{T}, \mathcal{B}(\mathbb{T}), \Pi)$, we shall use the convention of saying that events in the prior space are almost surely true (a.s.; with respect to $\omega$), and that events in the latter space occur almost everywhere (a.e.; with respect to $\theta$), when the events occur with probability 1 in the respective spaces.

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

A1 For each $\theta \in \mathbb{T}$, $D(X_n, Y_n) \xrightarrow{n \to \infty} D_{\infty}(\theta_0, \theta)$.

A2 For each $\epsilon > 0$, $\sup_{\delta \in \mathbb{R}_{\ge 0}} w(\delta, \epsilon) < \infty$.

A3 For each $\epsilon > 0$, $w(D_{\infty}(\theta_0, \cdot), \epsilon) : \mathbb{T} \to \mathbb{R}_{\ge 0}$ is continuous on a set $\tilde{\mathbb{T}} \subset \mathbb{T}$, such that $\Pi(\tilde{\mathbb{T}}) = 1$.

**THEOREM 1.** Under A1–A3, if

$$ \int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) > 0, $$

then (i) for almost every $\theta \in \mathbb{T}$:

$$ \bar{\pi}_{\epsilon,n}(\theta) \xrightarrow{n \to \infty} \pi_{\epsilon}(\theta) = \frac{\pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon)}{\int_{\mathbb{T}} \pi(\tau) w(D_{\infty}(\theta_0, \tau), \epsilon) \mathrm{d}\mathbf{n}(\tau)}, $$

and (ii) for almost every $\theta \in \mathbb{T}$:

$$ \pi_{\epsilon}(\theta | \mathbf{X}_{n}) \xrightarrow{n \to \infty} \pi_{\epsilon}(\theta). $$

Let us first 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. Notice that no assumption is made regarding the independence between elements of the sequences $X_{\infty}$ and $Y_{\infty}$. Further, no direct conditions are imposed on the limiting function $D_{\infty}$, 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.

**Remark 1.** Theorem 1(i) can be viewed as a generalization of the large-sample results regarding the coarsened-posterior, proved in Miller & Dunson (2019, Sec. S3.1), extended to cases where $X_{\infty}$ is a stochastic sequence with non-degenerate measure. Theorem 1(ii) makes the same conclusion as Jiang (2018, Thm. 1) and Nguyen et al. (2020, Thm. 2), although the proofs in the aforementioned works require some clarification, which we take the opportunity to make. All proofs of main results can be found in the Appendix of this text, in the Supplementary Materials.

To proceed, we require the notion of *almost sure weak convergence*, which is defined in Berti et al. (2006) and Grübel & Kabluchko (2016). Let $\mathcal{P}(\mathbb{T})$ be the space of probability measures on $\mathbb{T}$, and let $(\Pi_{\omega,n})_{n \in \mathbb{N}}$ be a sequence measures in $\mathcal{P}(\mathbb{T})$, indexed by $\omega \in \Omega$. We say that $\Pi_{\omega,n}$ almost surely weakly converges to $\Pi \in \mathcal{P}(\mathbb{T})$, if there exists a set $\tilde{\Omega} \in \mathcal{F}$ such that $P(\tilde{\Omega}) = 1$ and for every $\omega \in \tilde{\Omega}$, $\Pi_{\omega,n}$ converges weakly to $\Pi$, as $n \to \infty$. By the usual definition of weak convergence, this is equivalent to the condition that

$$ \int_{\mathbb{T}} f(\theta) \mathrm{d}\Pi_{\omega,n}(\theta) \xrightarrow{n \to \infty} \int_{\mathbb{T}} f(\theta) \mathrm{d}\Pi(\theta), $$

for every bounded and continuous function $f : \mathbb{T} \to \mathbb{R}$, for each $\omega \in \tilde{\Omega}$.
② $\epsilon \to 0$ and $n \to \infty$: when the tolerance parameter and the sample size tend to zero and infinity, respectively, the goal is to have a more stringent acceptance criterion with a large sample, potentially leading to more accurate estimates of the posterior distribution, with high probability or in expectation. One typically requires shape restrictions on the tail probabilities of both the measure on the sequence of data and the prior measure. These restrictions can be controlled as a function of $\epsilon$ and are often quite restrictive and intricate. The results that have been obtained in this setting yield convergence in probability theorems which differ from our conclusions, which establish convergence of measures in the almost sure sense. An archetypal example of such a result is found in Bernton et al. (2019), Proposition 3 and Corollary 1.

③ $\epsilon \to 0$ and $n$ fixed: when the tolerance parameter decreases toward zero while the sample size is kept constant, the acceptance criterion becomes strict, requiring that whatever distance is used to measure similarities can identify discretely supported measures. It has been resolved for most practical scenarios through Bernton et al. (2019), Proposition 2, and Rubio and Johansen (2013), Proposition 1.

A collection of recent works from the literature concerning these asymptotic settings for ABC posterior measures appears in Table 1.

<table><caption>Table 1: Recent works on the asymptotics of ABC posterior measures, organized by the behavior of $\epsilon$ and $n$. The top and bottom rows correspond to pseudo-posteriors and coarsened posteriors, respectively.</caption><thead><tr><th>① &epsilon; fixed and n &rarr; &infin;</th><th>② &epsilon; &rarr; 0 and n &rarr; &infin;</th><th>③ &epsilon; &rarr; 0 and n fixed</th></tr></thead><tbody><tr><td>Jiang (2018)</td><td>Bernton et al. (2019)</td><td>Bernton et al. (2019)</td></tr><tr><td>Nguyen et al. (2020)</td><td>Frazier et al. (2018)</td><td>Rubio and Johansen (2013)</td></tr><tr><td>Legramanti et al. (2025)</td><td>Li and Fearnhead (2018a,b)</td><td>Nguyen et al. (2020)</td></tr><tr><td></td><td>Frazier et al. (2020)</td><td></td></tr><tr><td></td><td>Legramanti et al. (2025)</td><td></td></tr><tr><td>Miller and Dunson (2019)</td><td></td><td>Miller and Dunson (2019)</td></tr></tbody></table>

The current work extends upon the best-known results for setting ① when $\epsilon > 0$ and $n \to \infty$, and seeks to complement the technical contributions by other authors regarding alternative settings for establishing convergence. Namely, in Section 3, we prove the almost sure convergence of the pseudo-posterior density and its convergence in expectation, resulting in a special case of the convergence for the coarsened posterior of Miller and Dunson (2019). Our convergence result makes more explicit assumptions than the previous works of Jiang (2018), Miller and Dunson (2019), and Nguyen et al. (2020), with proofs that more carefully handle the measure theoretic elements of the analysis. Furthermore, we make the important observation that these posterior density results imply the convergence in total variation of the corresponding measures. In Section 4, we then expand upon the convergence of the ABC posterior measures (both the pseudo-posterior and coarsened posterior measures) to obtain broadly applicable concentration
COROLLARY 1. Let the measures $\Pi_{\epsilon,n}$ and $\bar{\Pi}_{\epsilon,n}$ be defined by

$$ \Pi_{\epsilon,n}(\mathbb{A}) = \int_{\mathbb{A}} \pi_{\epsilon}(\theta | \mathbf{X}_{n}) \, \mathrm{d}\mathbf{n}(\theta), \quad \text{and} \quad \bar{\Pi}_{\epsilon,n}(\mathbb{A}) = \int_{\mathbb{A}} \bar{\pi}_{\epsilon,n}(\theta) \, \mathrm{d}\mathbf{n}(\theta), $$

for $\mathbb{A} \in \mathcal{B}(\mathbb{T})$, respectively, for each $n \in \mathbb{N}$. Then, under A1-A3, (i) $\bar{\Pi}_{\epsilon,n}$ converges weakly to $\Pi_{\epsilon}$, and under A1-A3, (ii) $\Pi_{\epsilon,n}$ converges almost surely weakly to $\Pi_{\epsilon}$, where $\Pi_{\epsilon}$ is defined by

$$ \Pi_{\epsilon}(\mathbb{A}) = \int_{\mathbb{A}} \pi_{\epsilon}(\theta) \, \mathrm{d}\mathbf{n}(\theta). $$

### 3. CONCENTRATION OF MASS

We let $\mathcal{B}_{\theta} = \{\tau \in \mathbb{R}^q : \|\theta - \tau\|_2 < 1\}$ be the (open) unit ball in $\mathbb{R}^q$ with respect to the Euclidean norm $\|\cdot\|_2$, centered at $\theta \in \mathbb{R}^q$, and we note that we can scale $\mathcal{B}_0$ by a factor $\lambda > 0$ to obtain balls $\lambda\mathcal{B}_0 = \{\lambda b : b \in \mathcal{B}_0\}$ with any radius $\lambda$. Let $T_0 \subset \mathbb{T}$ be some set of interest. We say that $\Theta_{\lambda} \subset \mathbb{T}$ is the set of the $\lambda$-covering centres for $T_0$ if $T_0 \subset \Theta_{\lambda} \oplus \lambda\mathcal{B}_0$, where $\mathbb{A} \oplus \mathcal{B} = \{a + b : a \in \mathbb{A}, b \in \mathcal{B}\}$ is the Minkowski sum. If $\Theta_{\lambda}$ is a finite set, then we will denote its cardinality by $|\Theta_{\lambda}|$. We shall denote the Lebesgue measure on $\mathbb{R}^q$ by Leb.

Let $\theta_0 \in \mathbb{T}$ and make the following assumptions:

B1 Let $D_{\infty}(\theta_0, \cdot) : \mathbb{T} \to \mathbb{R}_{\ge 0}$ and assume that the set of its zeroes $\Theta_0 = \{\theta \in \mathbb{T} : D_{\infty}(\theta_0, \theta) = 0\}$ is non-empty.

B2 For each $\lambda > 0$, there exists an $\epsilon > 0$, such that if $D_{\infty}(\theta_0, \theta) < \epsilon$, then $\inf_{\tau \in \Theta_0} \|\theta - \tau\|_2 < \lambda$.

B3 For each $\lambda > 0$, there exists a covering of $\Theta_0$ with $\lambda$-covering centres $\Theta_{\lambda} \subset \mathbb{T}$, such that $\lambda^q |\Theta_{\lambda}| \xrightarrow{\lambda \to 0} 0$.

B4 The weight function $w(\cdot, \cdot)$ can be decomposed as

$$ w(\delta, \epsilon) = W(\delta, \epsilon) \llbracket \varphi(\delta) < \epsilon \rrbracket, $$

where $W: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ and $\varphi: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is strictly increasing and bijective, with $\varphi(0) = 0$.

THEOREM 2. Assume B1-B3. Then, (i) as $\lambda \to 0$,

$$ \mathrm{Leb}(\Theta_0 \oplus \lambda \mathcal{B}_0) \to 0. $$

Let $\int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \, \mathrm{d}\mathbf{n}(\theta) > 0$, for each $\epsilon > 0$. If we further assume A1-A3 and B4, then (ii) for every $\lambda > 0$, there exists an $\epsilon > 0$, such that

$$ \bar{\Pi}_{\epsilon,n}(\Theta_0 \oplus \lambda \mathcal{B}_0) \xrightarrow{n \to \infty} 1, $$

and (iii) for every $\lambda > 0$, there exists an $\epsilon > 0$, such that

$$ \Pi_{\epsilon,n}(\Theta_0 \oplus \lambda \mathcal{B}_0) \xrightarrow{n \to \infty} 1. $$

Let us now unpack the assumptions and conclusions of Theorem 2. Firstly, B1 simply assumes that the zeroes of $D_{\infty}(\theta_0, \theta) = 0$ exists. If $D_{\infty}: \mathbb{T} \times \mathbb{T} \to \mathbb{R}_{\ge 0}$ is a pseudometric on $\mathbb{T}$ (as per Richmond 2020, Sec. 11.1), then we may consider $\Theta_0$ to be the equivalent class defined by the generative parameter: $\Theta_0 = [\theta_0]$, corresponding to the equivalence relationship: $D_{\infty}(\theta_0, \theta) = 0$. B2 is a primitive boundedness and an identification assumption. Here, if we again consider $D_{\infty}$ to be a pseudometric, then we may consider the definition of a new metric over equivalent
of mass results as $n \to \infty$, in practical cases when the density $p_{\theta_0}(\mathbf{x}_n)$ is not identifiable with respect to the generative parameter $\theta_0$.

**Outline** In the rest of the paper, we first discuss the related literature in Section 2, and then define notations as well as the pseudo-posterior density and the coarsened posterior density in Section 3. Our main results on convergence of ABC posterior measures and concentration of mass without identifiability are stated in Section 4 and Section 5, respectively. The next three sections are devoted to illustrations: posterior concentration is illustrated in Section 6, numerical estimation of the posterior is covered in Section 7, while Section 8 covers an application motivated by a sound source localization problem. Section 9 summarizes our contributions and discusses future research directions, while the Appendix (Nguyen et al., 2025) contains proofs of the technical results.

## 2 Related work

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 $\Pi_{n,\epsilon}^c$ and $\Pi_{n,\epsilon}^p$. 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 $L_1$ 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 $\Pi_{n,\epsilon}^c$ or $\Pi_{n,\epsilon}^p$ converges to a point mass, in some mode of convergence, when $\epsilon$ is taken to be a decreasing function of $n$, as $n \to \infty$. 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 $\mathbf{X}_n$ and $\mathbf{Y}_n$, particularly the concentration of the prior measure $\Pi$,
classes $T_\sim = \{[\theta] : \theta \in \mathbb{T}\}$ denoted by $D_\sim : T_\sim \times T_\sim \to \mathbb{R}_{\ge 0}$. Then B2 is satisfied by taking $D_\sim$ to be Hölder continuous with respect to the *Euclidean Hausdorff distance* on sets $\|\cdot - \cdot\|_\sim : T_\sim \times T_\sim \to \mathbb{R}_{\ge 0}$ defined by

$$ \|[\theta] - [\tau]\|_{\sim} = \max \left\{ \sup_{t_1 \in [\theta]} \inf_{t_2 \in [\tau]} \|t_1 - t_2\|_2, \sup_{t_1 \in [\tau]} \inf_{t_2 \in [\theta]} \|t_1 - t_2\|_2 \right\}, $$

in the sense that there exists an $L, K > 0$ such that $\|[\theta] - [\tau]\|_{\sim} \le LD_\sim ([\theta], [\tau])^K$, for every $\theta, \tau \in \mathbb{T}$.

Another simplification of B2 can be obtained by taking $\Theta_0 = \{\theta_0\}$ to be a singleton. Then, B2 is implied by the condition that $D_\infty$ is a metric and the identity function $\theta \mapsto \theta$ continuously maps the metric spaces $(\mathbb{T}, D_\infty)$ and $(\mathbb{T}, \|\cdot - \cdot\|_2)$, uniformly. This again is implied by the Lipschitz/Hölder condition that there exists an $L, K > 0$ such that $\|\theta - \tau\|_2 \le LD_\infty (\theta, \tau)^K$, for every $\theta, \tau \in \mathbb{T}$.

Next, B3 states that there exists a set of covering centres $\Theta_\lambda$ of $\Theta_0$ that does not grow too quickly, since we need the Lebesgue measure of the Euclidean balls covering $\Theta_0$ to be small when $\lambda$ is small, in order to establish Part (i) of the theorem. Here, the assumption is automatically fulfilled when $\Theta_0$ is a finite set, which particularly holds true when $\Theta_0$ is a singleton. Note, however that countability of $\Theta_0$ is not a sufficient condition for B3.

Lastly, B4 implies that $w(D_\infty(\theta_0, \cdot), \epsilon)$ and its support are bounded, when taken together with A2. This holds for the classical *accept/reject kernel* $w(\delta, \epsilon) = \|[\delta - \epsilon]\|$, but not for kernels with unbounded support, such as the Gaussian kernel $w(\delta, \epsilon) = \exp(-\delta^2/\epsilon)$, that is considered in Park et al. (2016) and Nguyen et al. (2020).

We can interpret Theorem 2 as follows. Part (i) states that under B1-B3, we can always cover the zero set $\Theta_0$, of elements of $\mathbb{T}$ that are indistinguishable from $\theta_0$, using $D_\infty$, 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, Part (ii) states that if we further assume A1-A3 and B4, we have the fact that given any choice of $\lambda > 0$, we can pick an $\epsilon > 0$ such that the ABC posterior measure, $\bar{\Pi}_{\epsilon,n}$ or $\Pi_{\epsilon,n}$, of the covering of $\Theta_0$ converges to full mass as $n \to \infty$. That is, regardless of how small the Lebesgue measure of our covering of $\Theta_0$ is, we can always choose an $\epsilon > 0$ such that the ABC posterior always eventually concentrates its mass entirely within the covering. Part (iii) then makes the equivalent conclusion regarding the pseudo-posterior $\Pi_{\epsilon,n}$, in the almost sure sense.

The conclusions can be viewed as a kind of *posterior consistency*, as defined in Ghosal & 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 $n \to \infty$, in some sense, in the case where $\Theta_0$ is a singleton. Here, we can call our conclusion a *nearly posterior consistency* result, since we obtain the fact that the ABC posterior measures concentrate on a sets of negligible mass, instead, for any size of negligibility, and for potentially uncountable $\Theta_0$.

**Remark 2.** Like Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2022), we can also permit misspecification between the density of the data generating process $p(x_n|\theta_0)$ and that of the simulated data $p(y_n|\theta)$, by allowing for the possibility that $D_\infty(\theta_0, \theta) > 0$, for all $\theta \in \mathbb{T}$. This can be achieved by replacing $\Theta_0$ in B1 by $\Theta_* = \{\theta \in \mathbb{T} : D_\infty(\theta_0, \theta) = \epsilon_*\}$, where $\epsilon_* = \min_{\theta \in \mathbb{T}} D_\infty(\theta_0, \theta)$ (assuming that $\epsilon_*$ exists), and replacing B2 by the condition: for each $\lambda > 0$, there exists an $\epsilon > 0$, such that if $D_\infty(\theta_0, \theta) < \epsilon + \epsilon_*$, then $\inf_{\tau \in \Theta_*} \|\theta - \tau\|_2 < \lambda$.
permit Frazier et al. (2018); Li and Fearnhead (2018b,a), and Frazier et al. (2020) to establish weak and strong convergence of $\Pi_{n,\epsilon}^p$ to measures with explicit forms.

## 3 Pseudo-posterior and coarsened posterior densities

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space with typical element $\omega$ and expectation operator $\mathrm{E}_\mathrm{P}$. We let $\mathbf{X}_\infty = (X_i)_{i \in \mathbb{N}}$ be a sequence of random variables $(X_i : \Omega \to \mathbb{X} \subset \mathbb{R}^d)$, and we denote the tuple containing the first $n$ elements of $\mathbf{X}_\infty$ by $\mathbf{X}_n$, which is defined on the measurable space $(\mathbb{X}^n, \mathcal{B}(\mathbb{X})^{\otimes n})$. We define a measure on $\mathbf{X}_n$ via the density function $p_{\theta_0}(\mathbf{x}_n)$, dominated by the measure $\mathbf{m}^{\otimes n}$ (where $\mathbf{m}$ is typically the counting measure or Lebesgue measure), for some $\theta_0 \in \mathbb{T}$, which we call the *generative parameter*.

Next, we let $\theta$ denote a random parameter from the measure space $(\mathbb{T}, \mathcal{B}(\mathbb{T}), \Pi)$, where $\Pi$ is characterized by its density $\pi$, dominated by the measure $\mathbf{n}$. Let $\mathbf{Y}_\infty = (Y_i)_{i \in \mathbb{N}}$ ($Y_i : \Omega \to \mathbb{X}$) be a sequence of random variables with partial sequence $\mathbf{Y}_n$ on $(\mathbb{X}^n, \mathcal{B}(\mathbb{X})^{\otimes n})$, and define the joint measure of $\mathbf{Y}_n$ and $\theta$ via the density $p(\mathbf{y}_n|\theta)\pi(\theta)$, where $p(\mathbf{y}_n|\theta)$ is the likelihood of $\mathbf{Y}_n$, and corresponds to a measure on $(\mathbb{X}^n, \mathcal{B}(\mathbb{X})^{\otimes n})$, for each $\theta \in \mathbb{T}$. Throughout the text, we will take $\mathbf{X}_\infty$ and $\mathbf{Y}_\infty$ to be independent, and take $\mathbb{X}$ and $\mathbb{T}$ to be Polish spaces. Furthermore, for fixed $\theta \in \mathbb{T}$, we will write $P_\theta$ 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 $\mathbf{X}_n$ and density $p(\mathbf{y}_n|\theta)$, for all $n \in \mathbb{N}$.

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

$$ \pi_{\epsilon,n}^{p}(\theta) = \pi_{\epsilon}^{p}(\theta|\mathbf{X}_{n}) = \frac{\pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_{n}|\theta)}{\mathcal{C}_{\epsilon}^{p}(\mathbf{X}_{n})}, \quad \epsilon \in \mathbb{R}_{> 0}, \qquad (1) $$

where

$$ \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) = \int_{\mathbb{X}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) p(\mathbf{y}_n|\theta) \mathrm{d}\mathbf{m}(\mathbf{y}_n), \\ \mathcal{C}_{\epsilon}^{p}(\mathbf{X}_n) = \int_{\mathbb{T}} \pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \mathrm{d}\mathbf{n}(\theta); $$

and the *coarsened posterior density* $\pi_{\epsilon,n}^c$ by

$$ \pi_{\epsilon,n}^{c}(\theta) = \frac{\pi(\theta) \mathrm{E}_{\mathrm{P}}\{\mathcal{L}_{\epsilon}(\mathbf{X}_{n}|\theta)\}}{\mathcal{C}_{\epsilon,n}^{c}}, \quad \epsilon \in \mathbb{R}_{> 0}, \qquad (2) $$

where

$$ \mathcal{C}_{\epsilon,n}^{c} = \int_{\mathbb{T}} \pi(\theta) \mathrm{E}_{\mathrm{P}}\{\mathcal{L}_{\epsilon}(\mathbf{X}_{n}|\theta)\} \mathrm{d}\mathbf{n}(\theta). $$
This then provides a result in situations when the underlying data generating process is both unidentifiable and misspecified.

## 4. ILLUSTRATIVE EXAMPLES OF POSTERIOR CONCENTRATION

### 4.1. Identifiable normal model

Let us suppose that $\mathbf{X}_{\infty}$ is an independent and identically distributed (IID) sequence defined by $X_i \sim N(\theta_0, 1)$ for each $i \in \mathbb{N}$, and that $\mathbf{Y}_{\infty}$ is independent of $\mathbf{X}_{\infty}$, with elements $Y_i \sim N(\theta, 1)$, where $\theta_0, \theta \in \mathbb{T} = \mathbb{R}$. We will take $\pi(\theta) = \phi(\theta; 0, 1)$, where $\phi(\cdot; \mu, \sigma^2)$ is the normal density function with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. We shall use the distance

$$D(\mathbf{X}_n, \mathbf{Y}_n) = \left| n^{-1} \sum_{i=1}^{n} X_i - n^{-1} \sum_{i=1}^{n} Y_i \right| \quad (3)$$

The law of large numbers and continuous mapping implies that $D(\mathbf{X}_n, \mathbf{Y}_n) \xrightarrow{n \to \infty}$ $D_{\infty}(\theta_0, \theta) = |\theta_0 - \theta|$. Provided that $w$ satisfies A2 and A3, then Theorem 1 implies that the coarsened-posterior and pseudo-posterior densities

$$\bar{\pi}_{\epsilon,n}(\theta) = \frac{\phi(\theta; 0, 1) \mathrm{E} \left\{ \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^n \phi(y_i; \theta, 1) \mathrm{d}\mathbf{y}_n \right\}}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \mathrm{E} \left\{ \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^n \phi(y_i; \tau, 1) \mathrm{d}\mathbf{y}_n \right\} \mathrm{d}\tau} \quad (4)$$

and

$$\pi_{\epsilon,n}(\theta) = \frac{\phi(\theta; 0, 1) \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^n \phi(y_i; \theta, 1) \mathrm{d}\mathbf{y}_n}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^n \phi(y_i; \tau, 1) \mathrm{d}\mathbf{y}_n \mathrm{d}\tau} \quad (5)$$

converge to

$$\pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) w(|\theta_0 - \theta|, \epsilon)}{\int_{\mathbb{R}} \phi(\tau; 0, 1) w(|\theta_0 - \tau|, \epsilon) \mathrm{d}\tau}, \quad (6)$$

where we write from now on dm and dn as d for the Lebesgue measure. To make the conclusions of Theorem 2, we require a choice of $w$ that satisfies B4. Two possibilities are the venerable accept/reject kernel and the triweight kernel:

$$w(\delta, \epsilon) = \left\{1 - (\delta/\epsilon)^2\right\}^3 \llbracket \delta < \epsilon \rrbracket,$$

which corresponds to a choice of $W(\delta, \epsilon) = \left\{1 - (\delta/\epsilon)^2\right\}^3$ and $\phi(\delta) = \delta$. In either case, since $\Theta_0$ is equal to the singleton $\{\theta_0\}$, it is then procedural to verify the remaining assumptions B1-B3 of Theorem 2, which then implies that the ABC posteriors can be made to concentrate on sets of arbitrarily small Lebesgue measure by making $\epsilon$ sufficiently small. We visualize the concentration of the limiting measure $\Pi_{\epsilon}$ in Figure 1, for both the cases of the accept/reject and triweight kernels.

### 4.2. Finitely unidentifiable normal model

We now instead suppose that $\mathbf{X}_{\infty}$ is an IID sequence defined by $X_i \sim N(\theta_0^2, 1)$ for each $i \in \mathbb{N}$, and that $\mathbf{Y}_{\infty}$ is independent of $\mathbf{X}_{\infty}$, with elements $Y_i \sim N(\theta^2, 1)$, where $\theta_0, \theta \in \mathbb{T} = \mathbb{R}$. We again use the distance (3), which almost surely converges to $D_{\infty}(\theta_0, \theta) = |\theta_0^2 - \theta^2|$.

For non-zero generative parameter $\theta_0$, B1 holds with $\Theta_0 = \{-\theta_0, \theta_0\}$. B2 holds since $|\theta_0^2 - \theta^2| < \epsilon$ implies that $\min_{\tau \in \{-\theta_0, \theta_0\}} \{|\tau - \theta|\} < \sqrt{\epsilon} = \lambda$. The verification of the remaining
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 $w(\delta, \epsilon) = \mathbf{1}_{[0,\epsilon)}(\delta)$, where we use the indicator function notation: $\mathbf{1}_A(x) = 1$ if $x \in A$, and $\mathbf{1}_A(x) = 0$ otherwise. Other popular choices of weights including the exponential kernel $w(\delta, \epsilon) = \exp(-|\delta|^\alpha/\epsilon)$ (with $\alpha > 0$) (Park et al., 2016), with Gaussian kernel special case, where $\alpha = 2$, and the Epanechnikov kernel $w(\delta, \epsilon) = (3/4)\{1 - (\delta/\epsilon)^2\}\mathbf{1}_{[0,\epsilon)}(\delta)$ (Beaumont et al., 2002). In Section 5, we also consider the tri-weight kernel $w(\delta, \epsilon) = \{1 - (\delta/\epsilon)^2\}^3\mathbf{1}_{[0,\epsilon)}(\delta)$ 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 $\theta_0 \in \mathbb{T}$ such that $\mathbf{X}_n$ has density $p(\mathbf{x}_n|\theta_0)$. We shall refer to both (1) and (2) as ABC posterior densities, when it is immaterial to distinguish between them.

# 4 Convergence of ABC posterior measures

Since we are operating on both the probability spaces $(\Omega, \mathcal{F}, \mathbb{P})$ and $(\mathbb{T}, \mathcal{B}(\mathbb{T}), \Pi)$, 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 $\omega \in \Omega$ by P-a.s. (P-almost sure/surely). Similarly, for an event that is true for almost all $\theta \in \mathbb{T}$, we write $\Pi$-a.a. ($\Pi$-almost all/always).

Let the pseudo-posterior and coarsened posterior measures $\Pi_{\epsilon,n}^P$ and $\Pi_{\epsilon,n}^c$ be defined, respectively, as

$$ \Pi_{\epsilon,n}^{P}(\mathbb{A}) = \int_{\mathbb{A}} \pi_{\epsilon}^{P}(\theta | \mathbf{X}_{n}) \mathrm{d}\mathbf{n}(\theta), \quad \Pi_{\epsilon,n}^{c}(\mathbb{A}) = \int_{\mathbb{A}} \pi_{\epsilon,n}^{c}(\theta) \mathrm{d}\mathbf{n}(\theta), $$

for each $\mathbb{A} \in \mathcal{B}(\mathbb{T})$ and $n \in \mathbb{N}$. We show in this section that $\Pi_{\epsilon,n}^P$ and $\Pi_{\epsilon,n}^c$ 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 $D_{\infty}(\cdot, \cdot) : \mathbb{T} \times \mathbb{T} \to \mathbb{R}_{\ge 0}$ and make the following assumptions:

**A1** For $\Pi$-a.a $\theta$, $D(\mathbf{X}_n, \mathbf{Y}_n) \xrightarrow{n \to \infty}^{P_{\theta}-\text{a.s.}} D_{\infty}(\theta_0, \theta)$.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Limiting posterior densities of form (6) using the accept/reject kernel (left) and triweight kernel (right), with $\theta_0 = 1$, are graphed for a range of values of 11 threshold values $\epsilon$, between 0.1 and 1. The prior densities are drawn with dotted lines.</figcaption></figure>

assumptions A1–A3 and B3–B4 for the case of $w$ set to the accept/reject and triweight kernels follows from analogous arguments to those made in Section 4.1. As such, we have the fact that the ABC posterior measures concentrate mass on sets with arbitrarily small Lebesgue measure, for sufficiently small $\epsilon > 0$, and have respective limiting densities:

$$ \pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) \left[ \left| \theta_0^2 - \theta^2 \right| < \epsilon \right]}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \left[ \left| \theta_0^2 - \tau^2 \right| < \epsilon \right] d\tau}, \quad (7) $$

and

$$ \pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) \left\{ 1 - \left( \frac{|\theta_0^2 - \theta^2|}{\epsilon} \right)^2 \right\}^3 \left[ \left| \theta_0^2 - \theta^2 \right| < \epsilon \right]}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \left\{ 1 - \left( \frac{|\theta_0^2 - \tau^2|}{\epsilon} \right)^2 \right\}^3 \left[ \left| \theta_0^2 - \tau^2 \right| < \epsilon \right] d\tau}, \quad (8) $$

when $w$ is taken to be either the accept/reject or triweight kernels. We plot the concentration of mass of the corresponding measures for various values of $\epsilon > 0$ in Figure 2.

### 4.3. Infinitely unidentifiable normal model

We further complicate the situation by supposing that $X_\infty$ is an IID sequence defined by $X_i \sim N(||\theta_0||_1, 1)$ for each $i \in \mathbb{N}$, and that $Y_\infty$ is independent of $X_\infty$, with IID elements $Y_i \sim N(||\theta||_1, 1)$. In this example, $\theta_0 = (\theta_{01}, \theta_{02})$ and $\theta = (\theta_1, \theta_2)$ are elements in $\mathbb{T} = \mathbb{R}^2$ with $||\theta_0||_1 = |\theta_{01}| + |\theta_{02}|$ and $||\theta||_1 = |\theta_1| + |\theta_2|$ denoting their $L_1$-norms. We naturally use the prior $\pi(\theta) = \phi(\theta_1; 0, 1) \phi(\theta_2; 0, 1)$ and consider the distance (3), which almost surely converges to $D_\infty(\theta_0, \theta) = |||\theta_0||_1 - |||\theta||_1|$. B1 holds with

$$ \Theta_0 = \{ \tau = (\tau_1, \tau_2) : ||\tau||_1 = ||\theta_0||_1 \} $$
**A2** For each $\epsilon > 0$, $\sup_{\delta \in \mathbb{R}_{\ge 0}} w(\delta, \epsilon) < \infty$.

**A3** For each $\epsilon > 0$, $w(D_{\infty}(\theta_0, \cdot), \epsilon) : \mathbb{T} \to \mathbb{R}_{\ge 0}$ is continuous $\Pi$-a.a.

**Proposition 1.** Under A1-A3, if

$$ \int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) > 0, $$

then (i) for $\Pi$-a.a $\theta$,

$$ \pi_{\epsilon}^{\mathrm{P}}(\theta | \mathbf{X}_n) \xrightarrow[n \to \infty]{\mathrm{P}-\mathrm{a.s.}} \pi_{\epsilon}(\theta) = \frac{\pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon)}{\int_{\mathbb{T}} \pi(\tau) w(D_{\infty}(\theta_0, \tau), \epsilon) \mathrm{d}\mathbf{n}(\tau)}, $$

and (ii) for $\Pi$-a.a $\theta$,

$$ \pi_{\epsilon,n}^{c}(\theta) \xrightarrow{n \to \infty} \pi_{\epsilon}(\theta). $$

*Proof sketch.* For each fixed $\theta \in \mathbb{T}$ for which A3 holds, we observe that

$$ w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon) \xrightarrow[n \to \infty]{\mathrm{P}_{\theta}-\mathrm{a.s.}} w(D_{\infty}(\theta_0, \theta), \epsilon) $$

which then allows the use of Hunt's Lemma (Lemma 1 in Appendix A (Nguyen et al., 2025)) to conclude that

$$ \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \xrightarrow[n \to \infty]{\mathrm{P}_{\theta}-\mathrm{a.s.}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \quad (3) $$

and

$$ \int_{\Omega} \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \mathrm{d}P_{\theta} \xrightarrow{n \to \infty} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) $$

for $\Pi$-a.a $\theta \in \mathbb{T}$, which yield the respective convergence of the numerators of (1) and (2). To obtain convergence of the denominator of (2), we apply Tonelli's Theorem to write $C_{\epsilon,n}^c$ as a double integral over the space $\Omega \times \mathbb{T}$:

$$ C_{\epsilon,n}^{c} = \int_{\Omega \times \mathbb{T}} \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \mathrm{d}Q(\omega, \theta), $$

where $Q$ is the joint measure on $(\Omega \times \mathbb{T}, \mathcal{F} \otimes \mathcal{B}(\mathbb{T}))$ that is consistent with $P$, $\Pi$, and $P_{\theta}$ (for each $\theta \in \mathbb{T}$), which exists by the Ionescu-Tulcea extension Theorem. This then permits an application of the dominated convergence theorem with respect to the joint measure $Q$ to show that $C_{\epsilon,n}^c$ converges to the integral of the right-hand side of (3).

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

$$ C_{\epsilon,n}^{\mathrm{P}}(\mathbf{X}_n) \xrightarrow[n \to \infty]{\mathrm{Q}-\mathrm{a.s.}} \int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) $$

by a second application of Hunt's Lemma.

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /><figcaption>Fig. 2. Limiting posterior densities (7) with accept/reject kernel (left) and (8) with triweight kernel (right), with $\theta_0 = 2$, are graphed for a range of 11 threshold values $\epsilon$, between 0.2 and 3. The prior densities are drawn with dotted lines.</figcaption></figure>

and B2 holds with $\epsilon = \lambda$ since $|||\theta_0||_1 - ||\theta||_1| < \epsilon$ implies that $\inf_{\tau \in \Theta_0} \{||\tau - \theta||_1\} < \epsilon = \lambda$ and $||\tau - \theta||_2 \le ||\tau - \theta||_1$ for each $\tau \in \Theta_0$. We also have to verify B3, which can be achieved by noting that $\Theta_0$ is the contour of the $L_1$-ball of radius $||\theta_0||_1$ and thus consists of four line segments, each of length $a\sqrt{2}$, where $a = ||\theta_0||_1$. We can cover each of line segments with Euclidean balls of radius $\lambda > 0$ by placing a ball on each of the $a\sqrt{2}/(2\lambda)$ equidistant points along the line, of distance $2\lambda$ apart. Then, we can check B3 by evaluating $\lambda^q |\Theta_\lambda| = 4 \times \lambda^2 (a\sqrt{2}/(2\lambda)) = 2a\lambda\sqrt{2}$, which approaches zero as $\lambda \to 0$, as required. The verification of A1–A3 and B4 for concluding Theorem 1 Part (i) and Theorem 2 Part (ii) using $w$ set to the accept/reject and triweight kernels is then procedural. We demonstrate concentration of mass by plotting the support of the limits of both of the ABC posterior measures, defined by (4) and (5), in Figure 3. For both kernels, the support satisfies $|||\theta||_1 - ||\theta_0||_1| \le \epsilon$ or equivalently $||\theta_0||_1 - \epsilon \le ||\theta||_1 \le ||\theta_0||_1 + \epsilon$, which corresponds to the region between the $L_1$-balls of radii $||\theta_0||_1 + \epsilon$ and $||\theta_0||_1 - \epsilon$.

**Remark 3.** Besides our choices for $w$, one can make use of any number of common kernels with compact supports from the theory of density estimation, such as the triangular or Epanechnikov kernels (cf. Scott 2015, Ch. 6).

**Remark 4.** We have used the example of normal distributions for the measures of $X_i$, $Y_i$ ($i \in \mathbb{N}$), and $\theta$, in all of the examples above out of convenience. Of course, the same illustrations can be made if we replace all uses of the normal law $N(\mu, \sigma^2)$, with density $\phi(\cdot; \mu, \sigma^2)$, by any generic location-scale law defined by density $\psi(\cdot; \mu, \sigma)$ with location and scale parameters $\mu \in \mathbb{R}$ and $\sigma > 0$, provided that the necessary integrals with respect to $\psi$ exist.

# 5. NUMERICAL ESTIMATION OF POSTERIOR MEASURES

We have opted for examples with summary-based discrepancy functions $D$ and for IID sequences $X_\infty$ and $Y_\infty$, for ease of understanding and simplicity of exposition. However, the
Complete proofs of all results can be found in Appendix B (Nguyen et al., 2025).

Let us now discuss the assumptions. A1 holds in a variety of settings and is gen-
erally obtained by combining a strong law of large numbers and continuous map-
ping theorem. As applied in Section 5, a common situation that often appears (see,
e.g., Marin et al., 2012) is when $D(\mathbf{X}_n, \mathbf{Y}_n) = \mathcal{D}(\varsigma_n(\mathbf{X}_n), \varsigma_n(\mathbf{Y}_n))$, where, for each
$n \in \mathbb{N}$, $\varsigma_n : \mathbb{X}^n \to \mathcal{S}$ is a summary statistic that maps to the space $\mathcal{S}$, and $\mathcal{D} :$
$\mathcal{S} \times \mathcal{S} \to \mathbb{R}_{\ge 0}$ is continuous in both its arguments. Then, if the summary statistics
are consistent in the sense that $\varsigma_n(\mathbf{X}_n) \xrightarrow[n \to \infty]{\text{P-a.s.}} \varsigma_0$ and $\varsigma_n(\mathbf{Y}_n) \xrightarrow[n \to \infty]{\text{P}_{\theta-a.s.}} \varsigma_{\theta}$, for $\Pi$-a.a $\theta$,
then A1 holds with $D_{\infty}(\theta_0, \theta) = \mathcal{D}(\varsigma_{\theta_0}, \varsigma_{\theta})$. In Section 6 we also consider the case
when $D(\mathbf{X}_n, \mathbf{Y}_n) = \int_{\mathbb{R}} |\bar{F}_n(x; \mathbf{X}_n) - F_n(x; \mathbf{Y}_n)| dx$, where $\bar{F}_n(\cdot; \mathbf{X}_n) : \mathbb{R} \to [0, 1]$
is the empirical distribution function corresponding to the sample $\mathbf{X}_n$ of observa-
tions supported on $\mathbb{X} = \mathbb{R}$. Then, by the Glivenko-Cantelli Theorem, it holds that
$\bar{F}_n(\cdot; \mathbf{X}_n)$ converges uniformly P-a.s to the distribution $F_{\theta_0}$ and similarly, $\bar{F}_n(\cdot; \mathbf{Y}_n)$
converges uniformly $\text{P}_{\theta}$-a.s to $F_{\theta}$. This then implies that $D(\mathbf{X}_n, \mathbf{Y}_n) \xrightarrow[n \to \infty]{\text{P}_{\theta-a.s.}} D_{\infty}(\theta_0, \theta)$,
where $D_{\infty}(\theta_0, \theta) = \int_{\mathbb{R}} |F_{\theta_0}(x) - F_{\theta}(x)| dx$, by Makarov and Podkorytov (2013, Thm.
4.8.1). Beyond these examples, we have that A1 holds, under appropriate conditions,
when $D(\mathbf{X}_n, \mathbf{Y}_n)$ 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**_∞ and **Y**_∞. Further, no direct conditions are imposed on the limiting
function *D*<sub>∞</sub>, 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.

Remark 1. Proposition 1(i) makes the same conclusion as Jiang (2018, Thm. 1) and Nguyen et al. (2020, Thm. 2), although the proofs in the aforementioned works require some clarification, which we take the opportunity to make. Proposition 1(ii) can be viewed as a version of the large-sample results regarding the well-specified case of the coarsened posterior, proved in Miller and Dunson (2019, Section S3.1).

Remark 2. The pointwise convergence established by Proposition 1, together with Scheffé’s Theorem, implies that

$$
\sup_{A \in \mathcal{B}(\mathcal{T})} \left| \Pi_{\epsilon,n}^{\text{P}}(\mathbb{A}) - \Pi_{\epsilon}(\mathbb{A}) \right| \xrightarrow[n \to \infty]{\text{P-a.s.}} 0, \text{ and } \sup_{A \in \mathcal{B}(\mathcal{T})} \left| \Pi_{\epsilon,n}^{\text{c}}(\mathbb{A}) - \Pi_{\epsilon}(\mathbb{A}) \right| \xrightarrow[n \to \infty]{\text{P}_{\theta-a.s.}} 0,
$$

where

$$
\Pi_{\epsilon}(\mathbb{A}) = \int_{\mathbb{A}} \pi_{\epsilon}(\theta) \, \mathrm{d}\mathbf{n}(\theta) \qquad (4)
$$

defines the limiting ABC posterior measure. That is, $\Pi_{\epsilon,n}^{\text{P}}$ and $\Pi_{\epsilon,n}^{\text{c}}$ converge in total variation to $\Pi_{\epsilon}$, almost surely and deterministically, respectively.

*Remark 3.* The conclusions of Proposition 1 hold irrespective of the identifiability of the model. Theorem 1 in the next section provides more precise results in terms of concentration of mass under specific treatments of the potential model unidentifiability.
<figure><img src="image_4.png" /><figcaption>Fig. 3. Supports of limiting posterior measures from Section 4.3 for various threshold values $\epsilon$, when $\theta_0 = (1, 1)$ and $||\theta_0||_1 = 2$.</figcaption></figure>

main conclusions of Theorems 1 and 2 apply in much broader settings, as we will demonstrate below.

### 5.1. Non-IID sequences from a first-order autoregressive model

In this example, we consider $X_\infty = (X_i)_{i \in \mathbb{N}}$, where $X_i$ can be characterized as

$$X_i = |\theta_0| X_{i-1} + E_i,$$

where $\theta_0 \in (-1, 1) = \mathbb{T}$, $X_0 \sim N(0, 1/\{1 - |\theta_0|^2\})$, and $(E_i)_{i \in \mathbb{N}}$ is IID with $E_i \sim N(0, 1)$. Similarly, we write $Y_\infty = (Y_i)_{i \in \mathbb{N}}$, characterized by

$$Y_i = |\theta_i| Y_{i-1} + E'_i,$$

with $\theta \in \mathbb{T}$, where $Y_0 \sim N(0, 1/\{1 - |\theta|^2\})$ and $(E'_i)_{i \in \mathbb{N}}$ is IID with $E'_i \sim N(0, 1)$. We shall use a uniform prior density $\pi(\theta) = \frac{\|\theta\| < 1}{2}$. Via Hall & Heyde (1980, Thm. 6.6), we have the following strongly consistent estimators of $|\theta_0|$ and $|\theta|$ from $X_n$ and $Y_n$: $r_{0,n} = \sum_{i=2}^n X_i X_{i-1}/\sum_{i=2}^n X_i^2$ and $r_n = \sum_{i=2}^n Y_i Y_{i-1}/\sum_{i=2}^n Y_i^2$, respectively. Thus, the distance $D(X_n, Y_n) = |r_{0,n} - r_n|$ converges almost surely to $D_\infty(\theta_0, \theta) = ||\theta_0| - |\theta||$.

To estimate the ABC posterior measures, we sample $(\theta_k)_{k \in [m]}$ and compute $(D_k)_{k \in [m]}$ as described in Algorithm 1. For any given $\epsilon > 0$, we can then characterize the ABC posterior measures via their distributions functions, estimated by their respective empirical distribution functions. The larger the value of $m$, the more accurate the empirical distribution is to its target.
# 5 Concentration of mass without identifiability

We let $\mathbb{B}_{\theta} = \{\tau \in \mathbb{R}^q : \|\theta - \tau\|_2 < 1\}$ be the (open) unit ball in $\mathbb{R}^q$ with respect to the Euclidean norm $\|\cdot\|_2$, centered at $\theta \in \mathbb{R}^q$, and we note that we can scale $\mathbb{B}_0$ by a factor $\lambda > 0$ to obtain balls $\lambda\mathbb{B}_0 = \{\lambda b : b \in \mathbb{B}_0\}$ with any radius $\lambda$. Let $T_0 \subset \mathbb{T} \subset \mathbb{R}^q$ be some set of interest. We say that $\Theta_\lambda \subset \mathbb{T}$ is a set of $\lambda$-covering centres for $T_0$ if $T_0 \subset \Theta_\lambda \oplus \lambda\mathbb{B}_0$, where $A \oplus B = \{a+b : a \in A, b \in B\}$ is the Minkowski sum. We will denote the cardinality of $\Theta_\lambda$ by $|\Theta_\lambda|$ and the Lebesgue measure on $\mathbb{R}^q$ by Leb.

Let $\theta_0 \in \mathbb{T}$. In the context of potential unidentifiability, consider the function $D_\infty(\theta_0, \cdot) : \mathbb{T} \to \mathbb{R}_{\ge 0}$ and denote its set of zeros by $\Theta_0 = \{\theta \in \mathbb{T} : D_\infty(\theta_0, \theta) = 0\}$. Then unidentifiability amounts to having a set of zeros $\Theta_0$ of cardinality larger than 1. The illustrations of Section 6 will cover the identifiable case $|\Theta_0| = 1$, and then the unidentifiable cases $1 < |\Theta_0| < \infty$ and $|\Theta_0| = \infty$.

We make the following assumptions:

**B1** The set of zeros of $D_\infty(\theta_0, \cdot) : \mathbb{T} \to \mathbb{R}_{\ge 0}$, denoted by $\Theta_0$, is non-empty.

**B2** For each $\lambda > 0$, there exists an $\epsilon > 0$, such that $D_\infty(\theta_0, \theta) < \epsilon$ implies that $\inf_{\tau \in \Theta_0} \|\theta - \tau\|_2 < \lambda$.

**B3** For each $\lambda > 0$, there exists a covering of $\Theta_0$ with $\lambda$-covering centres $\Theta_\lambda \subset \mathbb{T}$, such that $\lambda^q |\Theta_\lambda| \xrightarrow{\lambda \to 0} 0$.

**B4** The weight function $w(\cdot, \cdot)$ can be decomposed as $w(\delta, \epsilon) = W(\delta, \epsilon) \mathbf{1}_{[0,\epsilon)}(\varphi(\delta))$ where $W: \mathbb{R}_{\ge 0} \times \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ and $\varphi: \mathbb{R}_{\ge 0} \to \mathbb{R}_{\ge 0}$ is strictly increasing and bijective, with $\varphi(0) = 0$.

**Lemma 1.** Assume B1-B3. Then, as $\lambda \to 0$, Leb($\Theta_0 \oplus \lambda\mathbb{B}_0$) $\to 0$.

*Proof sketch.* We make the observation that $\Theta_0 \oplus \lambda\mathbb{B}_0 \subset \Theta_\lambda \oplus 2\lambda\mathbb{B}_0$, and combine this with the fact that the Lebesgue measure of $\lambda\mathbb{B}_0 \le c\lambda^q$ in $\mathbb{R}^q$, for some constant $c > 0$. The result follows by bounding Leb($\Theta_\lambda \oplus 2\lambda\mathbb{B}_0$) from above by $|\Theta_\lambda|$ Leb($2\lambda\mathbb{B}_0$). ■

**Theorem 1.** Assume A1-A3 and B1-B4. Let $\int_{\mathbb{T}} \pi(\theta) w(D_\infty(\theta_0, \theta), \epsilon) \, dn(\theta) > 0$, for each $\epsilon > 0$. Then

(i) for every $\lambda > 0$, there exists an $\epsilon > 0$, such that $\Pi_{\epsilon,n}^p(\Theta_0 \oplus \lambda\mathbb{B}_0) \xrightarrow{n \to \infty} 1$;

(ii) for every $\lambda > 0$, there exists an $\epsilon > 0$, such that $\Pi_{\epsilon,n}^c(\Theta_0 \oplus \lambda\mathbb{B}_0) \xrightarrow{n \to \infty} 1$.

*Proof sketch.* Let $S_0^\epsilon \subset \mathbb{T}$ be the support of the ABC posterior kernel in the sense that $w(D_\theta(\theta_0, \theta), \epsilon) = 0$ for all $\theta \notin S_0^\epsilon$. By B4, this set has a finite radius in the sense that for every $\epsilon > 0$, $D_\infty(\theta_0, \theta) < \varphi^{-1}(\epsilon)$ if $\theta \in S_0^\epsilon$. From this, B1 and B2 then imply that $S_0^\epsilon \subset \Theta_0 \oplus \lambda\mathbb{B}_0$, which implies that the mass of the limiting posterior measure (4) is concentrated on $\Theta_0 \oplus \lambda\mathbb{B}_0$, since B4 implies that $\Pi_\epsilon(\Theta_0 \oplus \lambda\mathbb{B}_0) \ge \Pi_\epsilon(S_0^\epsilon) = \Pi_\epsilon(\mathbb{T}) = 1$.
When we take $w$ to be the accept/reject kernel, the empirical distribution function is given by:

$$F_{m, \epsilon}^{\text{AR}}(\theta) = \left\{ \sum_{k=1}^{m} \llbracket D_k < \epsilon \rrbracket \right\}^{-1} \sum_{k=1}^{m} \llbracket D_k < \epsilon, \theta_k < \theta \rrbracket. \quad (9)$$

Similarly, when taking $w$ to be the triweight kernel, we can estimate distribution functions that characterize the ABC posterior measures, for any given $\epsilon$, via the empirical weighted distribution function:

$$F_{m, \epsilon}^{\text{tri}}(\theta) = \left\{ \sum_{k=1}^{m} \left\{ 1 - \left( \frac{D_k}{\epsilon} \right)^2 \right\}^3 \llbracket D_k < \epsilon \rrbracket \right\}^{-1} \sum_{k=1}^{m} \left\{ 1 - \left( \frac{D_k}{\epsilon} \right)^2 \right\}^3 \llbracket D_k < \epsilon, \theta_k < \theta \rrbracket. \quad (10)$$

Figure 4 displays sample functions (9) and (10) from an experiment with $\theta_0 = 1/2$, $n \in \{100, 1000\}$, and $m = 10000$. We observe that the empirical distribution functions both concentrate around the estimates $\{-r_{0,n}, r_{0,n}\}$, which converge towards the zeroes of $D_\infty(\theta_0, \theta)$: $\Theta_0 = \{-\theta_0, \theta_0\} = \{-1/2, 1/2\}$. The support of the sample measures are supersets of the estimates $\{-r_{0,n}, r_{0,n}\}$ that decrease in size as $\epsilon$ decreases towards zero, as expected. We also observe that, for fixed $\epsilon$ the empirical distribution functions both converge to their limiting forms, as $n$ increases, as predicted by Corollary 1.

**Algorithm 1** Monte Carlo sampling for estimation of the pseudo-/coarsened-posterior measures.

**Input:** Data $\mathbf{X}_n$, discrepancy function $\bar{D}$, number of Monte Carlo replications $m \ge 1$.

**For** $k \in [m]$:

Sample $\theta_k$ from a measure with density $\pi(\theta)$;

Generate $\mathbf{Y}_{n,k}$ from a measure with density $f(\mathbf{y}_n|\theta_k)$;

Compute discrepancy $D_k = D(\mathbf{X}_n, \mathbf{Y}_{n,k})$.

**Output:** Discrepancies $\mathbf{D}_m = (D_k)_{k \in [m]}$; Parameters $\mathbf{\Theta}_m = (\theta_k)_{k \in [m]}$.

## 5.2. Wasserstein distance

For this example, we take $\mathbf{X}_\infty$ to be an IID sequence of random variables, where $X_i \sim N(||\theta_0||_2^2, 1)$, for each $i \in \mathbb{N}$. We let $\theta_0 = (\theta_{01}, \theta_{02}) \in \mathbb{T} = [-2, 2]^2$ and endow $\mathbb{T}$ with the prior measure defined by the uniform density $\pi(\theta) = \frac{[\theta_1, \theta_2] \in [-2, 2]^2}{16}$. We then take $\mathbf{Y}_\infty$ to be an IID sequence, independent of $\mathbf{X}_\infty$, such that $Y_i \sim N(||\theta||_2^2, 1)$, for each $i \in \mathbb{N}$.

To measure the distance between partial sequences $\mathbf{X}_n$ and $\mathbf{Y}_n$, we use the sample 1-Wasserstein distance

$$D(\mathbf{X}_n, \mathbf{Y}_n) = n^{-1} \sum_{i=1}^{n} |X_{(i)} - Y_{(i)}|,$$

where $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ and $Y_{(1)} \le Y_{(2)} \le \dots \le Y_{(n)}$ are the order statistics of $\mathbf{X}_n$ and $\mathbf{Y}_n$, respectively (cf. Peyré & Cuturi 2019, Rem. 2.28). In this case, the 1-Wasserstein distance between measures on $\mathbb{R}$ is just the $L_1$ distance between the distribution functions of $X_1$ and $Y_1$, $F_{\theta_0}$ and $F_\theta$ (say) (Peyré & Cuturi, 2019, Rem. 2.30). The Glivenko-Cantelli theorem implies that $D(\mathbf{X}_n, \mathbf{Y}_n)$ converges almost surely to $D_\infty(\theta_0, \theta) = \int_{-\infty}^\infty |F_{\theta_0}(x) - F_\theta(x)| dx$. To verify B2, we use the fact that the 1-Wasserstein distance between normal distributions $N(\mu_1, \sigma^2)$ and $N(\mu_2, \sigma^2)$ is $|\mu_1 - \mu_2|$, for $\mu_1, \mu_2 \in \mathbb{R}$ (cf. Chafai & Malrieu, 2010, Example. 2.5).
Then, A1–A3 permit the conclusions of Remark 2, implying the convergence in total variation of $\Pi_{\epsilon,n}^p$ and $\Pi_{\epsilon,n}^c$ to $\Pi_\epsilon$, P-a.s. and deterministically, respectively. We conclude by noting that convergence in total variation implies set-wise convergence. ■

Let us now unpack the assumptions and conclusions of Theorem 1. Firstly, B1 simply assumes that the zeroes of $D_\infty(\theta_0, \theta) = 0$ exists. If $D_\infty: \mathbb{T} \times \mathbb{T} \to \mathbb{R}_{\ge 0}$ is a pseudo-metric on $\mathbb{T}$ (as per Richmond 2020, Section 11.1), then we may consider $\Theta_0$ to be the equivalence class defined by the generative parameter: $\Theta_0 = [\theta_0]$, corresponding to the equivalence relationship: $D_\infty(\theta_0, \theta) = 0$.

B2 is a primitive boundedness and an identification assumption. A sufficient condition for B2 to hold is if the Hölder condition $\inf_{\tau \in \Theta_0} \|\theta - \tau\|_2 \le LD_\infty (\theta, \theta_0)^K$ is satisfied for each $\theta \in \mathbb{T}$, for some $L, K > 0$. This is further simplified when taking $\Theta_0 = \{\theta_0\}$ to be a singleton. Then, B2 is implied by condition that $\|\theta - \theta_0\|_2 \le LD_\infty (\theta, \theta_0)^K$, for every $\theta \in \mathbb{T}$.

Next, B3 states that there exists a set of covering centres $\Theta_\lambda$ of $\Theta_0$ that does not grow too quickly, since we need the Lebesgue measure of the Euclidean balls covering $\Theta_0$ to be small when $\lambda$ is small, in order to establish Part (i) of the theorem. Here, the assumption is automatically fulfilled when $\Theta_0$ is a finite set, which particularly holds true when $\Theta_0$ is a singleton. Note, however, that countability of $\Theta_0$ is not a sufficient condition for B3 as in the case of dense countable subsets of $\mathbb{R}^q$. Taken together, B1–B3 require that there is a non-empty set of parameters that $\Theta_0 \subset \mathbb{T}$ corresponding to the generative model of $X_n$, that the sets of parameters in $\mathbb{T}$ that produce small discrepancies to the generative model are good approximations of the set $\Theta_0$, and that the set $\Theta_0$ can be covered by sets of arbitrarily small volume, respectively.

Lastly, B4 implies, when taken together with A2, that $w(D_\infty(\theta_0, \cdot) \epsilon)$ and its support are bounded. This holds for the classical *accept/reject kernel* $w(\delta, \epsilon) = \mathbf{1}_{[0,\epsilon)}(\delta)$, but not for kernels with unbounded support, such as the Gaussian kernel $w(\delta, \epsilon) = \exp(-\delta^2/\epsilon)$, 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 $\Theta_0$, of elements of $\mathbb{T}$ that are indistinguishable from $\theta_0$, using $D_\infty$, 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 $\epsilon > 0$ such that the pseudo-posterior and coarsened posterior measures, $\Pi_{\epsilon,n}^c$ or $\Pi_{\epsilon,n}^p$, of the covering of $\Theta_0$ converge to full mass as $n \to \infty$, almost surely and deterministically, respectively. That is, regardless of how small the Lebesgue measure of our covering of $\Theta_0$ is, we can always choose an $\epsilon > 0$ 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
<figure><img src="image_5.png" /><figcaption>Fig. 4. Empirical distribution function $F_{m,\epsilon}^{\text{AR}}$ (9) (left) and weighted empirical distribution function $F_{m,\epsilon}^{\text{wi}}$ (10) (right) for the experiment described in Section 5.1, with $\theta_0 = 1/2$, $m = 10000$, and sample sizes equal to $n = 100$ (top row), $n = 1000$ (middle row), and $n = \infty$ (bottom row; limiting measures). Dotted diagonal lines represent the distribution function of the prior measure.</figcaption></figure>
posterior measure concentrates on a point mass (with zero Lebesgue measure, in the continuous case), as $n \to \infty$, in some sense, in the case where $\Theta_0$ 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 $\Theta_0$.

*Remark 4.* Like Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2025), we can also permit misspecification between the density of the data generating process of $\mathbf{X}_n$: $p_0(\mathbf{x}_n)$, and that of the simulated data $p(\mathbf{y}_n|\theta)$, in the sense that there does not exist a $\theta_0 \in \mathbb{T}$ such that $p_{\theta_0}(\mathbf{x}_n) = p(\mathbf{x}_n|\theta_0)$, by allowing for the possibility that $D_\infty(\theta_0, \theta) > 0$, for all $\theta \in \mathbb{T}$. This can be achieved by replacing $\Theta_0$ in B1 by $\Theta_* = \{\theta \in \mathbb{T}: D_\infty(\theta_0, \theta) = \epsilon_*\}$, where $\epsilon_* = \min_{\theta \in \mathbb{T}} D_\infty(\theta_0, \theta)$ (assuming that $\epsilon_*$ exists), and replacing B2 by the condition: for each $\lambda > 0$, there exists an $\epsilon > 0$, such that if $D_\infty(\theta_0, \theta) < \epsilon + \epsilon_*$, then $\inf_{\tau \in \Theta_*} \|\theta - \tau\|_2 < \lambda$. This then provides a result in situations when the underlying data generating process is both unidentifiable and misspecified.

# 6 Illustrative examples of posterior concentration

In this section we illustrate examples covering different sets of zeros $\Theta_0$. Firstly, in Section 6.1, $\Theta_0$ is equal to the singleton $\{\theta_0\}$, and then in Section 6.2, $\Theta_0$ is made of two points $\{\theta_0, -\theta_0\}$. Lastly, an example where $|\Theta_0| = \infty$ is studied in Section 6.3.

## 6.1 Identifiable normal model

Let us suppose that $\mathbf{X}_\infty$ is an independent and identically distributed (IID) sequence defined by $X_i \sim N(\theta_0, 1)$ for each $i \in \mathbb{N}$, and that $\mathbf{Y}_\infty$ is independent of $\mathbf{X}_\infty$, with elements $Y_i \sim N(\theta, 1)$, where $\theta_0, \theta \in \mathbb{T} = \mathbb{R}$. We will take $\pi(\theta) = \phi(\theta; 0, 1)$, where $\phi(\cdot; \mu, \sigma^2)$ is the normal density function with mean $\mu \in \mathbb{R}$ and variance $\sigma^2 > 0$. We shall use the distance

$$D(\mathbf{X}_n, \mathbf{Y}_n) = \left| n^{-1} \sum_{i=1}^{n} X_i - n^{-1} \sum_{i=1}^{n} Y_i \right| \quad (5)$$

The law of large numbers and continuous mapping implies that $D(\mathbf{X}_n, \mathbf{Y}_n) \xrightarrow{n \to \infty} D_\infty(\theta_0, \theta) = |\theta_0 - \theta|$, which entails that $\Theta_0$ is equal to the singleton $\{\theta_0\}$. Provided that $w$ satisfies A2 and A3, then Theorem 1 implies that the pseudo-posterior and coarsened posterior densities

$$\pi_{\epsilon,n}^{P}(\theta) = \frac{\phi(\theta; 0, 1) \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^{n} \phi(y_i; \theta, 1) d\mathbf{y}_n}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^{n} \phi(y_i; \tau, 1) d\mathbf{y}_n d\tau} \quad (6)$$

and

$$\pi_{\epsilon,n}^{c}(\theta) = \frac{\phi(\theta; 0, 1) \mathrm{E} \left\{ \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^{n} \phi(y_i; \theta, 1) d\mathbf{y}_n \right\}}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \mathrm{E} \left\{ \int_{\mathbb{R}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) \prod_{i=1}^{n} \phi(y_i; \tau, 1) d\mathbf{y}_n \right\} d\tau} \quad (7)$$
We again use Algorithm 1 to obtain estimates of the ABC posteriors using the accept/reject kernel and the triweight kernel in place of $w$. Simulations are carried out with $\theta_{01} = \theta_{02} = 1/\sqrt{2}$, $n \in \{100, 1000\}$, and $m = 10000$. Interpreting the expression $\theta_k < \theta$ as meaning $\theta_{k1} < \theta_1$ and $\theta_{k2} < \theta_2$ (for $\theta_k, \theta \in \mathbb{R}^2$), we provide representations of the obtained estimated empirical distribution functions of form (9) and (10) via their weighted point masses in Figure 5, along with an illustration of the supports of the limiting ABC posterior measures for various values of $\epsilon$. We observe, as predicted, that the estimated ABC posterior measures concentrate on supersets of the zeroes of $D_\infty$, which can be characterized by the contour of the $L_2$-ball of radius $\|\theta_0\|_2 = 1$: $\Theta_0 = \{\theta = (\theta_1, \theta_2) \in \mathbb{R}^2 : \|\theta\|_2 = \|\theta_0\|_2\}$, where the sets get smaller in volume as $\epsilon$ decreases, and where smaller weights $w(D_k, \epsilon)$ are assigned for larger deviations from $\Theta_0$. As expected from Corollary 1, we observe that the supports of the empirical representations converge to those of the limiting measures, as $n$ increases, for fixed values of $\epsilon$.

To complement these numerical results, we provide an additional example in the Appendix, motivated by a sound source localization application from Forbes et al. (2022). Further, we note that when $X_\infty$ and $Y_\infty$ arise from general location-scale distributions instead of normal distributions, we can use the formula of Gelbrich (1990, Cor. 2.4), together with the Wasserstein norm equivalence result of Garling (2018, Cor. 21.2.4), in order to verify B2.

## 6. DISCUSSION

Investigations regarding the asymptotics of ABC algorithms have seen ongoing progress and our work is only one piece of a larger tapestry of findings that provide theoretical results and guarantees for such techniques. As such, we find that a discussion of the placement of our work in proximity to other treatments is useful and imperative. Firstly, we note that our work follows the progress of Jiang (2018), Miller & Dunson (2019), and Nguyen et al. (2020), who focus on the use of limit theorems to derive asymptotics regarding the ABC posteriors $\bar{\Pi}_{n,\epsilon}$ and $\Pi_{n,\epsilon}$. This differs from the approaches of Frazier et al. (2018), Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2022), 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 $L_1$ and almost sure convergence of posterior objects instead.

It is also notable that Theorem 2 only provides nearly posterior consistency results, rather than true posterior consistency, as discussed in Section 3, which would require a result that guarantees that $\bar{\Pi}_{n,\epsilon}$ or $\Pi_{n,\epsilon}$ converges to a point mass, in some mode of convergence, when $\epsilon$ is taken to be a decreasing function of $n$, as $n \to \infty$. Such a result is in fact guaranteed by Frazier et al. (2018, Thm. 1), Bernton et al. (2019, Cor. 1), and Legramanti et al. (2022, Cor. 1), in probability. We note that both Frazier et al. (2018, Thm. 1), Bernton et al. (2019, Cor. 1), and Legramanti et al. (2022, Cor. 1) make Hölder continuity assumptions that are stronger than our uniform continuity assumption when interpreting B2 in the identifiable case, and obviously much stronger than our assumptions in non-identifiable cases. In particular, in comparison to Frazier et al. (2018, Thm. 1), we do not require that $D$ be a distance between summary statistics. This is also true when comparing with the results of Li & Fearnhead (2018b,a).

Next, unlike Frazier et al. (2018), Bernton et al. (2019), Frazier et al. (2020), and Legramanti et al. (2022), we provide asymptotic results for weights $w$ other than the accept/reject kernel. This is a feature that is shared with Li & 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 & Fearnhead (2018b,a). However, the additional restrictions together with stronger assumptions regarding the concentration of measure and weak convergence of
<figure><img src="image_3.png" /><figcaption>Figure 1: Identifiable normal model. Limiting posterior densities of form (8) using the accept/reject kernel (left) and triweight kernel (right), with $\theta_0 = 1$, are graphed for a range of values of 11 threshold values $\epsilon$, between 0.1 and 0.6. The prior densities are drawn with dotted lines.</figcaption></figure>

converge to

$$ \pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) w(|\theta_0 - \theta|, \epsilon)}{\int_{\mathbb{R}} \phi(\tau; 0, 1) w(|\theta_0 - \tau|, \epsilon) d\tau}, \quad (8) $$

where we write from now on $dm$ and $dn$ as $d$ for the Lebesgue measure. To make the conclusions of Theorem 2, we require a choice of $w$ that satisfies B4. Two possibilities are the venerable accept/reject kernel and the triweight kernel $w(\delta, \epsilon) = \{1 - (\delta/\epsilon)^2\}^3 \mathbf{1}_{[0,\epsilon)}(\delta)$ which corresponds to a choice of $W(\delta, \epsilon) = \{1 - (\delta/\epsilon)^2\}^3$ and $\phi(\delta) = \delta$. In either case, since $\Theta_0 = \{\theta_0\}$, it is then procedural to verify the remaining assumptions B1–B3 of Theorem 2, which then implies that the pseudo-posterior and coarsened posterior can be made to concentrate on sets of arbitrarily small Lebesgue measure by making $\epsilon$ sufficiently small. We visualize the concentration of the limiting measure $\Pi_{\epsilon}$ in Figure 1, for both the cases of the accept/reject and triweight kernels. Observe that the masses of both limiting measures concentrate on smaller intervals around $\theta_0$ as $\epsilon$ decreases, as expected from our theory. The choice of weight functions influences the smoothness and shape of the posterior densities but does not affect the size of the support.

Remark 5. Besides our choices for $w$, one can use any common kernel with compact support from the theory of density estimation, such as the triangular or Epanechnikov kernels (cf. Scott 2015, Ch. 6).

## 6.2 Finitely unidentifiable normal model

We now instead suppose that $X_{\infty}$ is an IID sequence defined by $X_i \sim N(\theta_0^2, 1)$ for each $i \in \mathbb{N}$, and that $Y_{\infty}$ is independent of $X_{\infty}$, with elements $Y_i \sim N(\theta^2, 1)$, where $\theta_0, \theta \in \mathbb{T} = \mathbb{R}$. We again use the distance (5), which P-a.s. converges to $D_{\infty}(\theta_0, \theta) = |\theta_0^2 - \theta^2|$.

For non-zero generative parameter $\theta_0$, B1 holds with $\Theta_0 = \{-\theta_0, \theta_0\}$. B2 holds since $|\theta_0^2 - \theta^2| < \epsilon$ implies that $\min_{\tau \in \{-\theta_0, \theta_0\}} \{|\tau - \theta|\} < \sqrt{\epsilon} = \lambda$. The verification of the
<figure><img src="image_6.png" /><figcaption>Fig. 5. Point mass representations of empirical distribution function $F_{m,\epsilon}^{\text{AR}}$ (9) (left) and weighted empirical distribution function $F_{m,\epsilon}^{\text{tri}}$ (10) (right) for the experiment described in Section 5.2, with $\theta_0 = (1/\sqrt{2}, 1/\sqrt{2})$, $m = 10000$, sample size values: $n = 100$ (top row), $n = 1000$ (middle row), and $n = \infty$ (bottom row; support of limiting measures). Dotted lines represent the set $\Theta_0$. Left: point masses for different threshold values $\epsilon > 0$. Right: point masses for $\epsilon = 1$, with the sizes of points indicating the weights $w(D_k, \epsilon)$ (as a fraction of the maximum observed weight).</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 2: Finitely unidentifiable normal model. Limiting posterior densities (9) with accept/reject kernel (left) and (10) with triweight kernel (right), with $\theta_0 = 2$, are graphed for a range of 11 threshold values $\epsilon$, between 0.5 and 3. The prior densities are drawn with dotted lines.</figcaption></figure>

remaining assumptions A1–A3 and B3–B4 for the case of $w$ set to the accept/reject and triweight kernels follows from analogous arguments to those made in Section 6.1. As such, we have the fact that the pseudo-posterior and coarsened posterior measures concentrate mass on sets with arbitrarily small Lebesgue measure, for sufficiently small $\epsilon > 0$, and have limiting densities

$$ \pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) \mathbf{1}_{[0, \epsilon)}(|\theta_0^2 - \theta^2|)}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \mathbf{1}_{[0, \epsilon)}(|\theta_0^2 - \tau^2|) d\tau}, \quad (9) $$

and

$$ \pi_{\epsilon}(\theta) = \frac{\phi(\theta; 0, 1) \left\{ 1 - \frac{(\theta_0^2 - \theta^2)^2}{\epsilon^2} \right\}^3 \mathbf{1}_{[0, \epsilon)}(|\theta_0^2 - \theta^2|)}{\int_{\mathbb{R}} \phi(\tau; 0, 1) \left\{ 1 - \frac{(\theta_0^2 - \theta^2)^2}{\epsilon^2} \right\}^3 \mathbf{1}_{[0, \epsilon)}(|\theta_0^2 - \tau^2|) d\tau}, \quad (10) $$

when $w$ is taken as either the accept/reject or triweight kernels, respectively. We plot the concentration of mass of the corresponding measures for various values of $\epsilon > 0$ in Figure 2. Unlike Figure 1, we observe that the limiting posterior distributions not only concentrate mass around $\theta_0$ but also around $-\theta_0$, since $N(\theta_0^2, 1)$ is the same in both cases. As predicted by our theory, as $\epsilon$ decreases, the limiting distributions are supported on smaller sets containing $\theta_0$ and $-\theta_0$, which are independent of the choice of weights.

## 6.3 Infinitely unidentifiable normal model

We further complicate the situation by supposing that $X_{\infty}$ is an IID sequence defined by $X_i \sim N(\|\theta_0\|_1, 1)$ for each $i \in \mathbb{N}$, and that $Y_{\infty}$ is independent of $X_{\infty}$, with IID elements $Y_i \sim N(\|\theta\|_1, 1)$. In this example, $\theta_0 = (\theta_{01}, \theta_{02})$ and $\theta = (\theta_1, \theta_2)$ are elements in $\mathbb{T} = \mathbb{R}^2$ with $\|\theta_0\|_1 = |\theta_{01}| + |\theta_{02}|$ and $\|\theta\|_1 = |\theta_1| + |\theta_2|$ denoting their $L_1$-norms. We use the
summaries of $X_n$ and $Y_n$, and concentration of the prior measure $\Pi$ permits Frazier et al. (2018); Li & Fearnhead (2018b,a), and Frazier et al. (2020) to establish weak and strong convergence of $\Pi_{n,\epsilon}$ to measures with explicit forms.

In summary, we have provided a set of results that permit the establishment of nearly posterior consistency of ABC posterior measures in the $L_1$ and almost sure sense, in scenarios that lack identifiability and when discrepancies are possibly not Hölder continuous, and when the sequences $X_\infty$ and $Y_\infty$ are non-IID. Our results provide valuable complements to the existing literature that deliver theoretical guarantees to situations that are not covered by previous works.

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<figure><img src="image_5.png" /><figcaption>Figure 3: **Infinitely unidentifiable normal model.** Contour plot representation of limiting posterior measures for four threshold values $\epsilon$ where the data-generating process only depends on $\theta_0$ through $\|\theta_0\|_1$. The black square represents $\theta_0 = (1,1)$. Black dotted lines represent the set $\Theta_0$, where $\|\theta_0\|_1 = 2$.</figcaption></figure>

prior $\pi(\theta) = \phi(\theta_1; 0, 1) \phi(\theta_2; 0, 1)$ and consider the distance (5), which P-a.s. converges to $D_\infty(\theta_0, \theta) = \|\theta_0\|_1 - \|\theta\|_1$. B1 holds with $\Theta_0 = \{\tau = (\tau_1, \tau_2) : \|\tau\|_1 = \|\theta_0\|_1\}$ and B2 holds with $\epsilon = \lambda$ since $\|\theta_0\|_1 - \|\theta\|_1 < \epsilon$ implies that $\inf_{\tau \in \Theta_0} \{\|\tau - \theta\|_1\} < \epsilon = \lambda$ and $\|\tau - \theta\|_2 \le \|\tau - \theta\|_1$ for each $\tau \in \Theta_0$. We also have to verify B3, which can be achieved by noting that $\Theta_0$ is the contour of the $l_1$-ball of radius $\|\theta_0\|_1$ and thus consists of four line segments, each of length $a\sqrt{2}$, where $a = \|\theta_0\|_1$. We can cover each of line segments with Euclidean balls of radius $\lambda > 0$ by placing a ball on each of the $a\sqrt{2}/(2\lambda)$ equidistant points along the line, of distance $2\lambda$ apart. Then, we can check B3 by evaluating $\lambda^q |\Theta_\lambda| = 4 \times \lambda^2(a\sqrt{2}/(2\lambda)) = 2a\lambda\sqrt{2}$, which approaches zero as $\lambda \to 0$, as required. The verification of A1–A3 and B4 for concluding Theorem 1 Part (i) and Theorem 2 Part (ii) using $w$ set to the accept/reject and triweight kernels is then procedural. We illustrate the concentration of mass by plotting the support of the limit (8) of both of the ABC posterior densities, defined by (6) and (7), in Figure 3. For both kernels, the support satisfies $\|\theta\|_1 - \|\theta_0\|_1 \le \epsilon$ or equivalently $\|\theta_0\|_1 - \epsilon \le \|\theta\|_1 \le \|\theta_0\|_1 + \epsilon$, which corresponds to the region between the $L_1$-balls of radii $\|\theta_0\|_1 + \epsilon$ and $\|\theta_0\|_1 - \epsilon$.

**Remark 6.** We have used the example of normal distributions for the measures of $X_i, Y_i$ ($i \in \mathbb{N}$), and $\theta$, in all of the examples above out of convenience. Of course, the same illustrations can be made if we replace all uses of the normal law $N(\mu, \sigma^2)$, with density $\phi(\cdot; \mu, \sigma^2)$, by any generic location-scale law defined by density $\psi(\cdot; \mu, \sigma)$ with location and scale parameters $\mu \in \mathbb{R}$ and $\sigma > 0$, provided that the necessary integrals with respect to $\psi$ exist.
# 7 Numerical estimation of posterior measures

We have opted for examples with summary-based discrepancy functions $D$ and for IID sequences $\mathbf{X}_\infty$ and $\mathbf{Y}_\infty$, 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.

## 7.1 Non-IID sequences from a first-order autoregressive model

In this example, we consider $\mathbf{X}_\infty = (X_i)_{i \in \mathbb{N}}$, where $X_i$ can be characterized as

$$X_i = |\theta_0| X_{i-1} + E_i,$$

where $\theta_0 \in (-1, 1) = \mathbb{T}$, $X_0 \sim N(0, 1/\{1 - |\theta_0|^2\})$, and $(E_i)_{i \in \mathbb{N}}$ is IID with $E_i \sim N(0, 1)$. Similarly, we write $\mathbf{Y}_\infty = (Y_i)_{i \in \mathbb{N}}$, characterized by $Y_i = |\theta|Y_{i-1} + E'_i$, with $\theta \in \mathbb{T}$, where $Y_0 \sim N(0, 1/\{1 - |\theta|^2\})$ and $(E'_i)_{i \in \mathbb{N}}$ is IID with $E'_i \sim N(0, 1)$. We shall use a uniform prior density $\pi(\theta) = 1_{[0,1]}(|\theta|)/2$. Via Hall and Heyde (1980, Thm. 6.6), we have the following strongly consistent estimators of $|\theta_0|$ and $|\theta|$ from $\mathbf{X}_n$ and $\mathbf{Y}_n$: $r_{0,n} = \sum_{i=2}^n X_i X_{i-1}/\sum_{i=2}^n X_i^2$ and $r_n = \sum_{i=2}^n Y_i Y_{i-1}/\sum_{i=2}^n Y_i^2$, respectively. Thus, the distance $D(\mathbf{X}_n, \mathbf{Y}_n) = |r_{0,n} - r_n|$ converges, P-a.s., to $D_\infty(\theta_0, \theta) = ||\theta_0| - |\theta||$.

To estimate the ABC posterior measures, we sample $(\theta_k)_{k \in [m]}$ and compute $(D_k)_{k \in [m]}$ as described in Algorithm 1. For any given $\epsilon > 0$, we can then characterize the ABC posterior measures via their cumulative distribution functions (CDFs), estimated by the respective empirical CDFs. The larger the value of $m$, the more accurate the empirical distribution is to its target. When we take $w$ to be the accept/reject kernel, the empirical CDF is given by:

$$F_{m, \epsilon}^{\text{AR}}(\theta) = \left\{ \sum_{k=1}^{m} \mathbf{1}_{[0, \epsilon)}(D_k) \right\}^{-1} \sum_{k=1}^{m} \mathbf{1}_{[0, \theta)}(D_k) \mathbf{1}_{[0, \theta)}(\theta_k). \quad (11)$$

In the same way, when taking $w$ to be the triweight kernel, we can estimate CDFs that characterize the ABC posterior measures, for any given $\epsilon$, via the empirical weighted CDF:

$$F_{m, \epsilon}^{\text{tri}}(\theta) = \left\{ \sum_{k=1}^{m} \left\{ 1 - \left( \frac{D_k}{\epsilon} \right)^2 \right\}^3 \mathbf{1}_{[0, \epsilon)}(D_k) \right\}^{-1} \sum_{k=1}^{m} \left\{ 1 - \left( \frac{D_k}{\epsilon} \right)^2 \right\}^3 \mathbf{1}_{[0, \epsilon)}(D_k) \mathbf{1}_{[0, \theta)}(\theta_k). \quad (12)$$

Figure 4 displays sample functions (11) and (12) from an experiment with $\theta_0 = 1/2$, $n \in \{100, 1000\}$, and $m = 10000$. We observe that the empirical cumulative distribution functions both concentrate around the estimates $\{-r_{0,n}, r_{0,n}\}$, which converge towards the zeroes of $D_\infty(\theta_0, \theta)$: $\Theta_0 = \{-\theta_0, \theta_0\} = \{-1/2, 1/2\}$. The support of the sample measures are supersets of the estimates $\{-r_{0,n}, r_{0,n}\}$ that decrease in size as $\epsilon$ decreases towards zero, as expected. We also observe that, for fixed $\epsilon$, the empirical cumulative distribution functions both converge to their limiting forms, as $n$ increases, as predicted by Remark 2.
# Supplementary Materials for “Concentration results for approximate Bayesian computation without identifiability”

BY H. D. NGUYEN

School of Mathematics and Physics, Univ. of Queensland St.
Lucia 4067, Queensland Australia.
h.nguyen7@uq.edu.au

T. T. NGUYEN, J. ARBEL, AND F. FORBES

Univ. Grenoble Alpes, Inria, CNRS
Grenoble INP, LJK 38000 Grenoble, France.
trung-tin.nguyen@inria.fr julyan.arbel@inria.fr florence.forbes@inria.fr

## SUMMARY

This supplement contains an Appendix comprising proofs of the main results from the primary document. We also include external technical results required within the proofs. An additional numerical illustration is also reported in the final section.

## APPENDIX

### Technical results

**LEMMA 1.** Let $(\mathcal{F}_n)_{n \in \mathbb{N}}$ be a filtration of $\mathcal{F}$, and let $\mathcal{F}_\infty = \sigma (\bigcup_{n \in \mathbb{N}} \mathcal{F}_n)$. Let $(U_n)_{n \in \mathbb{N}}$ be a sequence of random variables, such that $U_n \xrightarrow{n \to \infty} U$, where $|U_n| \le V$, for each $n \in \mathbb{N}$, such that $E(V) < \infty$. Then, $\lim_{n \to \infty} E(U_n|\mathcal{F}_n) = E(U|\mathcal{F}_\infty)$, almost surely and in $L_1$.

**Remark 1.** Lemma 1 is often called Hunt's Lemma and can be found in Dellacherie & Meyer (1980) and Spataru (2013, Thm. 29.32).

### Proofs of main results

#### Proof of Theorem 1

We prove this result via Lemma 1. Start by setting

$$U_n = w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon),$$

where both $\mathbf{X}_n$ and $\mathbf{Y}_n$ are functions of $\omega \in \Omega$. By A2, we can set the (deterministic) constant $v = \sup_\delta w(\delta, \epsilon) < \infty$, so that $|U_n| \le v < \infty$. By A1

$$D(\mathbf{X}_n, \mathbf{Y}_n) \xrightarrow{n \to \infty} D_\infty(\theta_0, \theta)$$

and $w(\cdot, \epsilon)$ is almost surely continuous at $D_\infty(\theta_0, \theta)$ by A3, so by continuous mapping:

$$U_n \to U = w(D_\infty(\theta_0, \theta), \epsilon).$$

Now set $\mathcal{F}_n = \sigma(\mathbf{X}_n)$, which approaches $\mathcal{F}_\infty = \bigcup_{n \in \mathbb{N}} \mathcal{F}_n$. Lemma 1 applied directly implies that

$$E\{w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon) | \mathcal{F}_n\} \to E\{w(D_\infty(\theta_0, \theta), \epsilon) | \mathcal{F}_\infty\},$$
<figure><img src="image_6.png" /><figcaption>Figure 4: **Autoregressive model.** Pseudo-posterior CDF $F_{m,\epsilon}^{AR}$ (11) (left) and pseudo-posterior weighted CDF $F_{m,\epsilon}^{tri}$ (12) (right) for the experiment described in Section 7.1, with $\theta_0 = 1/2$, $m = 10000$, and sample sizes equal to (a) $n = 100$ (top row), (b) $n = 1000$ (middle row), and (c) $n = \infty$ (bottom row; limiting measures). Dotted diagonal lines represent the CDF of the prior measure (uniform).</figcaption></figure>
almost surely and in L₁, as $n \to \infty$. Note that the left-hand side reduces to

$$
\begin{align*}
E\{w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon)|\mathcal{F}_n\} &= E\{w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon)|X_1, \dots, X_n\} \\
&= \int_{\mathbb{X}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) p(\mathbf{y}_n|\theta) \mathrm{d}\mathbf{m}(\mathbf{y}_n) \\
&= \mathcal{L}_\epsilon(\mathbf{X}_n|\theta)
\end{align*}
$$

and that the right-hand side reduces to a constant, we have

$$
\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \underset{n \to \infty}{\xrightarrow{\text{a.s., } L_1}} w(D_{\infty}(\theta_0, \theta), \epsilon). \quad (1)
$$

To prove (i), we note that A3 implies that (1) holds for almost every $\theta \in \mathbb{T}$. The L₁ convergence
then implies that

$$
E\{\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)\} \xrightarrow{n \to \infty} w(D_{\infty}(\theta_0, \theta), \epsilon) \quad \text{and} \quad \pi(\theta) E\{\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)\} \xrightarrow{n \to \infty} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon).
$$

Applying the dominated convergence theorem, upon noting that $E\{\mathcal{L}_\epsilon(\mathbf{X}_n|\theta)\} \le \sup_\delta w(\delta, \epsilon)$ and that $\pi(\theta)$ is a density function, we then obtain

$$
\bar{C}_{\epsilon,n} \xrightarrow{n \to \infty} \int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \, \mathrm{d}n(\theta),
$$

which completes the proof.

To prove (ii), we again use A3 and (1) to now obtain the fact that for almost every $\theta \in \mathbb{T}$

$$
\pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \xrightarrow{n \to \infty} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon), \quad (2)
$$

almost surely. We now apply Lemma 1 in an alternative manner. That is, using the same filtration
($\mathcal{F}_n$)$_{n \in \mathbb{N}}$, we write the normalization term $\mathcal{C}_\epsilon(\mathbf{X}_n)$ as the conditional expectation

$$
E_{(\omega,\theta)}\{w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon)|\mathcal{F}_n\} = \int_{\mathbb{T}} \int_{\mathbb{X}^n} w(D(\mathbf{X}_n, \mathbf{y}_n), \epsilon) p(\mathbf{y}_n|\theta) \pi(\theta) \mathrm{d}\mathbf{m}(\mathbf{y}_n) \mathrm{d}\mathbf{n}(\theta)
$$

with respect to the product measure defined by the joint density $p(\mathbf{y}_n|\theta) \pi(\theta)$. By the same argument as used to obtain (1), Lemma 1 implies that

$$
E_{(\omega,\theta)}\{w(D(\mathbf{X}_n, \mathbf{Y}_n), \epsilon)|\mathcal{F}_n\} \to E_{(\omega,\theta)}\{w(D(\theta_0, \theta), \epsilon)|\mathcal{F}_\infty\}
$$

almost surely and in L₁, as $n \to \infty$. By Tonelli's theorem and by definition of $E_{(\omega,\theta)}$, we have

$$
\begin{align*}
E_{(\omega,\theta)}\{w(D(\theta_0, \theta), \epsilon) | \mathcal{F}_{\infty}\} &= \int_{\Omega} \int_{\mathbb{T}} w(D(\theta_0, \theta), \epsilon) \pi(\theta) \mathrm{d}\mathbf{n}(\theta) \mathrm{d}P(\omega) \\
&= \int_{\mathbb{T}} \pi(\theta) w(D(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta)
\end{align*}
$$

since $w(D(\theta_0, \theta), \epsilon)$ is bounded and does not depend on $\omega$. Thus,

$$
\mathcal{C}_{\epsilon}(\mathbf{X}_n) \xrightarrow{n \to \infty} \int_{\mathbb{T}} \pi(\theta) w(D(\theta_0, \theta), \epsilon) \mathrm{d}n(\theta). \quad (3)
$$

We then obtain the desired result via an application of Slutsky’s theorem for ratios.

**Proof of Corollary 1**
**Algorithm 1** Sampling from the prior and computing discrepancies.

**Input:** Data $\mathbf{X}_n$, discrepancy function $D$, number of Monte Carlo replications $m \ge 1$.

**For** $k \in [m]$:

Sample $\theta_k$ from a measure with density $\pi(\theta)$;

Generate $\mathbf{Y}_{n,k}$ from a measure with density $f(\mathbf{y}_n|\theta_k)$;

Compute discrepancy $D_k = D(\mathbf{X}_n, \mathbf{Y}_{n,k})$.

**Output:** Discrepancies $(D_k)_{k \in [m]}$; Parameters $(\theta_k)_{k \in [m]}$.

## 7.2 Wasserstein distance in Gaussian model

For this example, we take $\mathbf{X}_\infty$ to be an IID sequence of random variables, where $X_i \sim N(\|\theta_0\|_2^2, 1)$, for each $i \in \mathbb{N}$. We let $\theta_0 = (\theta_{01}, \theta_{02}) \in \mathbb{T} = [-2, 2]^2$ and endow $\mathbb{T}$ with the prior measure defined by the uniform density $\pi(\theta) = \mathbf{1}_{[-2,2]^2}(\theta_1, \theta_2)/16$. We then take $\mathbf{Y}_\infty$ to be an IID sequence, independent of $\mathbf{X}_\infty$, such that $Y_i \sim N(\|\theta\|_2^2, 1)$, for each $i \in \mathbb{N}$.

To measure the distance between partial sequences $\mathbf{X}_n$ and $\mathbf{Y}_n$, we use the sample 1-Wasserstein distance

$$D(\mathbf{X}_n, \mathbf{Y}_n) = n^{-1} \sum_{i=1}^{n} |X_{(i)} - Y_{(i)}|,$$

where $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ and $Y_{(1)} \le Y_{(2)} \le \dots \le Y_{(n)}$ are the order statistics of $\mathbf{X}_n$ and $\mathbf{Y}_n$, respectively (cf. Peyré and Cuturi 2019, Rem. 2.28). In this case, the 1-Wasserstein distance between measures on $\mathbb{R}$ is just the $L_1$ distance between the distribution functions of $X_1$ and $Y_1$, $F_{\theta_0}$ and $F_\theta$ (Peyré and Cuturi, 2019, Rem. 2.30). The Glivenko-Cantelli theorem implies that $D(\mathbf{X}_n, \mathbf{Y}_n)$ converges P-a.s. to $D_\infty(\theta_0, \theta) = \int_{-\infty}^\infty |F_{\theta_0}(x) - F_\theta(x)| dx$. To verify B2, we use the fact that the 1-Wasserstein distance between normal distributions $N(\mu_1, \sigma^2)$ and $N(\mu_2, \sigma^2)$ is $|\mu_1 - \mu_2|$, for $\mu_1, \mu_2 \in \mathbb{R}$ (cf. Chafai and Malrieu, 2010, Example. 2.5).

We use Algorithm 1 to obtain samples and discrepancies to estimate the ABC posteriors using the accept/reject kernel and the triweight kernel in place of $w$. Simulations are carried out with $\theta_{01} = \theta_{02} = 1/\sqrt{2}$, $n \in \{100, 1000\}$, and $m = 10000$. Interpreting the expression $\theta_k < \theta$ as meaning $\theta_{k1} < \theta_1$ and $\theta_{k2} < \theta_2$ (for $\theta_k, \theta \in \mathbb{R}^2$), we provide representations of the obtained estimated empirical distribution functions of form (11) and (12) via their weighted point masses in Figure 5, along with an illustration of the supports of the limiting ABC posterior measures for various values of $\epsilon$. We observe, as predicted, that the estimated ABC posterior measures concentrate on supersets of the zeroes of $D_\infty$, which can be characterized by the contour of the $L_2$-ball of radius $\|\theta_0\|_2 = 1$: $\Theta_0 = \{\theta = (\theta_1, \theta_2) \in \mathbb{R}^2 : \|\theta\|_2 = \|\theta_0\|_2 = 1\}$, where the sets get smaller in volume as $\epsilon$ decreases, and where smaller weights $w(D_k, \epsilon)$ are assigned for larger deviations from $\Theta_0$. As expected from Remark 2, we observe that the supports of the empirical representations converge to those of the limiting measures, as $n$ increases, for fixed values of $\epsilon$.
To prove Part (i), it is sufficient to check that

$$ \int_{\mathbb{T}} f(\theta) \pi(\theta) \mathrm{E}\{\mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta)\} \mathrm{d}\mathbf{n}(\theta) \xrightarrow{n \to \infty} \int_{\mathbb{T}} f(\theta) \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta), $$

for each bounded continuous function $f: \mathbb{T} \to \mathbb{R}$, which follows from the proof of Theorem 1 and an application of the dominated convergence theorem. Similarly, to prove Part (ii), it suffices to show that

$$ \int_{\mathbb{T}} f(\theta) \pi(\theta) \mathcal{L}_{\epsilon}(\mathbf{X}_n|\theta) \mathrm{d}\mathbf{n}(\theta) \xrightarrow{n \to \infty} \int_{\mathbb{T}} f(\theta) \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta), $$

which follows via the same argument used to obtain (3), with the additional fact that $f(\theta)$ is bounded and thus has finite expectation with respect to the product measure $P \times \Pi$.

### Proof of Theorem 2

We begin our proof of Part (i) by defining the Minkowski sum of sets $A, B \subset T$ as $A \oplus B = \{a+b : a \in A, b \in B\}$. By B3, we have $\lambda$-covering centres $\Theta_{\lambda} \subset T$ of $\Theta_0$ (as defined by B1 and B2). This then implies that $\Theta_0 \subset \Theta_{\lambda} \oplus \lambda B_0$. By the monotonicity of Minkowski addition, $\Theta_0 \oplus \lambda B_0 \subset \Theta_{\lambda} \oplus \lambda B_0 \oplus \lambda B_0 = \Theta_{\lambda} \oplus 2\lambda B_0$, where we use the fact that if $A$ is convex, then $\alpha A + \beta A = (\alpha + \beta) A$ (cf. Schneider 2013, Ch. 3). Now, using the fact that the Lebesgue measure of $\lambda B_0$ is $\pi^{q/2}\lambda^q/\Gamma(q/2+1)$ and by monotonicity, we obtain the fact that

$$ \mathrm{Leb}(\Theta_{\lambda} \oplus \lambda B_0) \le \mathrm{Leb}(\Theta_{\lambda} \oplus 2\lambda B_0) = |\Theta_{\lambda}| \frac{\pi^{q/2} (2\lambda)^q}{\Gamma(q/2+1)} \xrightarrow{\lambda \to 0} 0, $$

$$ \text{since } |\Theta_{\lambda}| \lambda^{q} \xrightarrow{\lambda \to 0} 0. $$

To prove Parts (ii) and (iii), notice that for each $\epsilon > 0$, $w(D_{\infty}(\theta_0, \theta), \epsilon)$ has support on $\{\theta : D_{\infty}(\theta_0, \theta) < \varphi^{-1}(\epsilon)\}$, by B4. But by B1 and B2, for each $\lambda > 0$, there exists a $\epsilon > 0$, such that if $D_{\infty}(\theta_0, \theta) < \varphi^{-1}(\epsilon)$, then $\min_{\tau \in \Theta_0} \|\theta - \tau\| < \lambda$, and thus

$$ \{\theta : D_{\infty}(\theta_0, \theta) < \varphi^{-1}(\epsilon)\} \subset \Theta_0 \oplus \lambda B_0, $$

by Part (i), and

$$ \int_{(\Theta_0 \oplus \lambda B_0) \cap T} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) \ge \int_{\{\varphi(D_{\infty}(\theta_0, \theta)) < \epsilon\} \cap T} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) \\ = \int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta), $$

where the equality holds by the fact that $w(D_{\infty}(\theta_0, \theta), \epsilon)$ is zero when $\varphi(D_{\infty}(\theta_0, \theta)) \ge \epsilon$. Thus,

$$ \frac{\int_{(\Theta_0 \oplus \lambda B_0) \cap T} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta)}{\int_{\mathbb{T}} \pi(\theta) w(D_{\infty}(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta)} = 1 $$
<figure><img src="image_7.png" /><figcaption>(a1) $n = 100$, $\epsilon$ varies</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(a2) $n = 100$, weights representation</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(b1) $n = 1000$, $\epsilon$ varies</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(b2) $n = 1000$, weights representation</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(c) $n = \infty$, $\epsilon$ varies</figcaption></figure>

Figure 5: **Student model.** Point mass representations (a1, b1) and contour plot (c) of pseudo-posterior distribution function for the accept-reject kernel, and pseudo-posterior weighted distribution function for the triweight kernel (a2, b2) for the experiment described in Section 7.2, with $m = 10000$ and where the black square represents $\theta_0 = (1/\sqrt{2}, 1/\sqrt{2})$. Black dotted lines represent the set $\Theta_0$. Left: point masses contributing to the empirical distribution under different threshold values $\epsilon > 0$ are indicated by color. Right: point masses for $\epsilon = 1$, with the sizes of points indicating the weights $w(D_k, \epsilon)$ (as a fraction of the maximum observed weight).
since $(\Theta_0 \oplus \lambda\mathbb{B}_0) \cap \mathbb{T} \subset \mathbb{T}$. Next, under A1-A3 we follow the proof of Corollary 1 Part (i), and conclude via the dominated convergence theorem that

$$ 
\begin{aligned}
\int_{(\Theta_0 \oplus \lambda\mathbb{B}_0) \cap \mathbb{T}} \pi(\theta) E\{\mathcal{L}_\epsilon(\mathbf{X}_n|\theta)\} \mathrm{d}\mathbf{n}(\theta) &= \int_{\mathbb{T}} [[\theta \in \Theta_0 \oplus \lambda\mathbb{B}_0]] \pi(\theta) E\{\mathcal{L}_\epsilon(\mathbf{X}_n|\theta)\} \mathrm{d}\mathbf{n}(\theta) \\
&\xrightarrow[n \to \infty]{} \int_{\mathbb{T}} [[\theta \in \Theta_0 \oplus \lambda\mathbb{B}_0]] \pi(\theta) w(D_\infty(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) \\
&= \int_{(\Theta_0 \oplus \lambda\mathbb{B}_0) \cap \mathbb{T}} \pi(\theta) w(D_\infty(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta),
\end{aligned}
$$

as required to show Part (ii). Similarly, Part (iii) follows from A1-A3, via the dominated convergence theorem on an almost sure set, to obtain

$$ 
\begin{aligned}
\int_{(\Theta_0 \oplus \lambda\mathbb{B}_0) \cap \mathbb{T}} \pi(\theta) \mathcal{L}_\epsilon(\mathbf{X}_n|\theta) \mathrm{d}\mathbf{n}(\theta) &= \int_{\mathbb{T}} [[\theta \in \Theta_0 \oplus \lambda\mathbb{B}_0]] \pi(\theta) \mathcal{L}_\epsilon(\mathbf{X}_n|\theta) \mathrm{d}\mathbf{n}(\theta) \\
&\xrightarrow[n \to \infty]{} \int_{\mathbb{T}} [[\theta \in \Theta_0 \oplus \lambda\mathbb{B}_0]] \pi(\theta) w(D_\infty(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta) \\
&= \int_{(\Theta_0 \oplus \lambda\mathbb{B}_0) \cap \mathbb{T}} \pi(\theta) w(D_\infty(\theta_0, \theta), \epsilon) \mathrm{d}\mathbf{n}(\theta),
\end{aligned}
$$

as in Corollary 1 Part (ii).

### Binaural sound source localisation

The following example is a simplified version of the synthetic sound source localization example of Forbes et al. (2022). Here, we take $\mathbf{X}_\infty$ to be an IID sequence of random variables, with $X_i \sim N(\mathrm{ITD}(\theta_0), 1/4)$, for each $i \in \mathbb{N}$, and $\theta_0 \in \mathbb{R}^2$ is interpreted as the 2D position of a sound source, assumed to be captured only through the noisy measurements of so-called interaural time differences (ITDs). In a binaural setting with two microphones, the ITD is defined as the difference between the time of arrival to the first and second microphone and given by $\mathrm{ITD}(\theta_0) = \|\|\theta_0 - \mu_1\|\|_2 - \|\theta_0 - \mu_2\|\|_2|$ for a source in $\theta_0$ and microphones at positions $\mu_1, \mu_2 \in \mathbb{R}^2$. ITD measurements only allow to determine a pair of hyperbolas on which the source maybe. We let $\theta_0 = (\theta_{01}, \theta_{02}) \in \mathbb{T} = [-2, 2]^2$ and endow $\mathbb{T}$ with the prior measure defined by the uniform density $\pi(\theta) = [(\theta_1, \theta_2) \in [-2, 2]^2]/16$. We then take $\mathbf{Y}_\infty$ to be an IID sequence, independent of $\mathbf{X}_\infty$, such that $Y_i \sim N(\mathrm{ITD}(\theta), 1/4)$, for each $i \in \mathbb{N}$.

We proceed in the same manner as in Section 5.2 of the main text. That is, to measure the distance between partial sequences $\mathbf{X}_n$ and $\mathbf{Y}_n$, we use the sample 1-Wasserstein distance

$$ D(\mathbf{X}_n, \mathbf{Y}_n) = n^{-1} \sum_{i=1}^{n} |X_{(i)} - Y_{(i)}|, $$

where $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ and $Y_{(1)} \le Y_{(2)} \le \dots \le Y_{(n)}$ are the order statistics of $\mathbf{X}_n$ and $\mathbf{Y}_n$, respectively. Again, the 1-Wasserstein distance between measures on $\mathbb{R}$ is just the $L_1$ distance between the distribution functions of $X_1$ and $Y_1$, $F_{\theta_0}$ and $F_\theta$, and we have the convergence of $D(\mathbf{X}_n, \mathbf{Y}_n)$, almost surely, to $D_\infty(\theta_0, \theta) = \int_{-\infty}^\infty |F_{\theta_0}(x) - F_\theta(x)| \mathrm{d}x$, with B2 verified again via the form of the 1-Wasserstein distance: $|\mathrm{ITD}(\theta_0) - \mathrm{ITD}(\theta)|$ (cf. Chafai & Malrieu, 2010, Example. 2.5).

We again use Algorithm 1 to obtain estimates of the ABC posteriors using the accept/reject kernel and the triweight kernel in place of $w$. Simulations are carried out with $\theta_{01} = \theta_{02} = 1$, $n \in \{100, 1000, 10000\}$, and $m = 10000$. We provide representations of the obtained estimated
Note that, when $X_\infty$ and $Y_\infty$ arise from general location-scale distributions instead of normal distributions, we can use the formula of Gelbrich (1990, Cor. 2.4) with the Wasserstein norm equivalence result of Garling (2018, Cor. 21.2.4) to verify B2.

## 8 Application to binaural sound source localization

The following example is a simplified version of a synthetic example of Forbes et al. (2022) related to sound source localization. Here, we take $X_\infty$ to be an IID sequence of random variables, with $X_i \sim N(\text{ITD}(\theta_0), 1/4)$, for each $i \in \mathbb{N}$, and $\theta_0 \in \mathbb{R}^2$ is interpreted as the 2D position of a sound source, assumed to be captured only through the noisy measurements of so-called interaural time differences (ITDs). In a binaural setting with two microphones, the ITD is defined as the difference between the time of arrival to the first and second microphone and given by $\text{ITD}(\theta_0) = \|\|\theta_0 - \mu_1\|_2 - \|\theta_0 - \mu_2\|_2|$ for a source in $\theta_0$ and microphones at positions $\mu_1, \mu_2 \in \mathbb{R}^2$. ITD measurements only allow us to determine a pair of hyperbolas on which the source may be. We let $\theta_0 = (\theta_{01}, \theta_{02}) \in \mathbb{T} = [-2, 2]^2$ and endow $\mathbb{T}$ with the prior measure defined by the uniform density $\pi(\theta) = \mathbf{1}_{[-2,2]^2}(\theta_1, \theta_2)/16$. We then take $Y_\infty$ to be an IID sequence, independent of $X_\infty$, such that $Y_i \sim N(\text{ITD}(\theta), 1/4)$, for each $i \in \mathbb{N}$.

We proceed in the same manner as in Section 7.2. That is, to measure the distance between partial sequences $X_n$ and $Y_n$, we use the 1-Wasserstein distance $D(\mathbf{X}_n, \mathbf{Y}_n) = n^{-1} \sum_{i=1}^n |X_{(i)} - Y_{(i)}|$, where $X_{(1)} \le X_{(2)} \le \dots \le X_{(n)}$ and $Y_{(1)} \le Y_{(2)} \le \dots \le Y_{(n)}$ are the order statistics of $X_n$ and $Y_n$, respectively. Again, the 1-Wasserstein distance between measures on $\mathbb{R}$ is just the $L_1$ distance between the distribution functions of $X_1$ and $Y_1$, $F_{\theta_0}$ and $F_{\theta}$, and we have the convergence of $D(\mathbf{X}_n, \mathbf{Y}_n)$, P-a.s., to $D_\infty(\theta_0, \theta) = \int_{-\infty}^\infty |F_{\theta_0}(x) - F_{\theta}(x)|dx$, with B2 verified via the form of the 1-Wasserstein distance: $|\text{ITD}(\theta_0) - \text{ITD}(\theta)|$ (cf. Chafai and Malrieu, 2010, Example. 2.5).

We again use Algorithm 1 to obtain prior and discrepancy samples to estimate the ABC posteriors using the accept/reject kernel and the triweight kernel in place of $w$. Simulations are carried out with $\theta_{01} = \theta_{02} = 1$, $n \in \{100, 1000, 10000\}$, and $m = 10000$. We provide representations of the obtained estimated empirical distribution functions via their weighted point masses in Figure 6. We observe, as predicted, that the estimated ABC posterior measures concentrate on supersets of the zeroes of $D_\infty$ which consists of two hyperbolas as shown in Figure 6. Observe that smaller values of $\epsilon$ correspond to sets with elements that sit closer to the zeroes of $D_\infty$. We also observe that smaller weights $w(D_k, \epsilon)$ are assigned for larger deviations from the zeroes of $D_\infty$.

## 9 Conclusion and future directions

In summary, we have provided a set of results that permit the establishment of posterior quasi-consistency of ABC posterior measures in the $L_1$ and almost sure sense, in scenarios that lack identifiability, when discrepancies are possibly not Hölder continuous, and when the sequences $X_\infty$ and $Y_\infty$ are potentially non-IID. Our results complement the existing literature and deliver theoretical guarantees to situations that are not covered by previous works.
empirical distribution functions via their weighted point masses in Figure 1, along with visual indicators of the supports of the limiting ABC posterior measures for various values of $\epsilon$. We observe, as predicted, that the estimated ABC posterior measures concentrate on supersets of the zeroes of $D_\infty$ which consists of two hyperbolas as shown in Figure 1. We visualize the boundaries of the support sets for different levels of $\epsilon$ via patterned lines on the plots. Observe that smaller values of $\epsilon$ correspond to sets with elements that sit closer to the zeroes of $D_\infty$. We also observe that smaller weights $w(D_k, \epsilon)$ are assigned for larger deviations from the zeroes of $D_\infty$. Furthermore, as expected, we observe that the supports of the empirical representations converge to those of the limiting measures, as $n$ increases, for each fixed value of $\epsilon$.

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<figure><img src="image_12.png" /><figcaption>(b1) $n = 1000$, $\epsilon$ varies</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(c1) $n = 10000$, $\epsilon$ varies</figcaption></figure>

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<figure><img src="image_17.png" /></figure>

**Figure 6: Sound source localization.** Point mass representations of pseudo-posterior distribution functions for the accept-reject kernel (left) and pseudo-posterior weighted distribution functions for the triweight kernel (right) for the binaural sound source localization experiment, with $\theta_0 = (1, 1)$, $m = 10000$. Black dots represent the locations of the two microphones. The black square represents the source $\theta_0$ on the thick-dotted hyperbola that visualizes the set $\Theta_0$. Left: point masses contributing to the empirical distribution under different threshold values $\epsilon > 0$, from 0.15 to 0.3, are indicated by color. Right: point masses for $\epsilon = 0.3$, with the sizes of points indicating the weights $w(D_k, \epsilon)$ (as a fraction of the maximum observed weight).
<figure><img src="image_7.png" /><figcaption>Fig. 1. Point mass representations of empirical distribution function $F_{m,\epsilon}^{AR}$ (9) (left) and weighted empirical distribution function $F_{m,\epsilon}^{tri}$ (10) (right) for the experiment described in the section labeled: Binaural sound source localisation, with $\theta_0 = (1, 1)$, $m = 10000$, sample size values: $n = 100$ (top row), $n = 1000$ (middle row), and $n = 10000$ (bottom row). Black dots represent the locations of the two microphones on the horizontal axis ($\mu_1$ and $\mu_2$), and another dot represents the source $\theta_0$ on the thick dotted hyperbola that visualizes the set $\Theta_0$. Additional patterned lines indicate the boundaries of the support of the limiting ABC posteriors for various fixed values of $\epsilon$. Left: point masses for different threshold values $\epsilon > 0$ from 0.15 to 0.3. Right: point masses for $\epsilon = 0.3$, with the sizes of points indicating the weights $w(D_k, \epsilon)$ (as a fraction of the maximum observed weight).</figcaption></figure>
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## Acknowledgments

We thank the Editor, Associate Editor, and anonymous Reviewers for constructive comments that markedly improved our manuscript.

## Funding

This work was supported by Australian Research Council grant number DP230100905 and Inria Project WOMBAT.

## Supplementary Material

Supplementary Material for “Revisiting concentration results for approximate Bayesian computation” (DOI: 10.1214/25-BA1520SUPP; .pdf). Appendix A contains some technical results and Appendix B contains proofs of the main results.

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