DeepAI

# On the use of ABC-MCMC with inflated tolerance and post-correction

Approximate Bayesian computation (ABC) allows for inference of complicated probabilistic models with intractable likelihoods using model simulations. The ABC Markov chain Monte Carlo (MCMC) inference is often sensitive to the tolerance parameter: low tolerance leads to poor mixing and large tolerance entails excess bias. We consider an approach using a relatively large tolerance for the ABC-MCMC to ensure sufficient mixing, and post-processing the output of ABC-MCMC leading to estimators for a range of finer tolerances. We introduce an approximate confidence interval for the related post-corrected estimators, which can be calculated from any run of ABC-MCMC, with little extra cost. We propose an adaptive ABC-MCMC, which finds a `balanced' tolerance level automatically, based on acceptance rate optimisation. Tolerance adaptation, combined with proposal covariance adaptation, leads to an easy-to-use adaptive ABC-MCMC, with subsequent post-correction over a range of tolerances. Our experiments show that post-processing based estimators can perform better than direct ABC-MCMC, that our confidence intervals are reliable, and that our adaptive ABC-MCMC leads to reliable inference with little user specification.

• 10 publications
• 4 publications
02/01/2019

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## 1. Introduction

Approximate Bayesian computation (ABC) is a form of likelihood-free inference (see, e.g., the reviews Marin et al., 2012; Sunnåker et al., 2013)

which is used when exact Bayesian inference of a parameter

with posterior density is impossible, where is the prior density and is an intractable likelihood with data . More specifically, when the generative model of observations cannot be evaluated, but allows for simulations, ABC can be used for relatively straightforward approximate inference, based on a pseudo-posterior

 (1) πϵ(θ)∝pr(θ)Lϵ(θ),whereLϵ(θ):=E[Kϵ(Yθ,y∗)],Yθ∼g(⋅∣θ),

where is a ‘tolerance’ parameter, and is a ‘kernel’ function, which is often taken as a simple cut-off , where

extracts a vector of summary statistics from the (pseudo) observations

The summary statistics are often chosen based on the application at hand, and reflect what is relevant for the inference task; see also (Fearnhead and Prangle, 2012). Because may be regarded as a smoothed version of the true likelihood using the kernel , it is intuitive that using a too large may blur the likelihood and bias the inference. Therefore, it is generally desirable to use as small a tolerance as possible, but because the computational ABC methods suffer from inefficiency with small , the choice of tolerance level is difficult (cf. Bortot et al., 2007; Sisson and Fan, 2018; Tanaka et al., 2006).

We discuss a simple post-processing procedure which allows for consideration of a range of values for the tolerance , based on a single run of ABC Markov chain Monte Carlo (ABC-MCMC) (Marjoram et al., 2003) with tolerance . Post-processing has been suggested earlier at least in (Wegmann et al., 2009) (in the special case of simple cut-off), and it can be regarded as an importance sampling correction of pseudo-marginal type MCMC (cf. Vihola et al., 2016). The method, discussed further in Section 2, can be useful for two reasons:

• A range of tolerances may be routinely inspected, which can reveal excess bias of ABC-MCMC with tolerance .

• The ABC-MCMC may be implemented with sufficiently large to allow for good mixing, and post-correction may be used for inference.

Our contribution is two-fold. We suggest straightforward-to-calculate approximate confidence intervals for the post-processing output, with some theoretical properties discussed in Section 3. We also introduce an adaptive ABC-MCMC in Section 4 which finds a balanced during burn-in, using acceptance rate as a proxy. We provide some experimental results regarding the suggested confidence interval and the tolerance adaptation in Section 5, and conclude with discussion in Section 6.

## 2. ABC-MCMC with post-processing over a range of tolerances

For the rest of the paper, we assume that the kernel function has the following form:

 Kϵ(y,y∗):=ϕ(d(y,y∗)ϵ),

where is any ‘dissimilarity’ function and is a non-increasing ‘cut-off’ function. Typically , where are the chosen summaries, and in case of the simple cut-off discussed in Section 1, . We also assume that the ABC posterior given in (1) is well-defined for all of interest (that is, ).

The following summarises the ABC-MCMC algorithm suggested by Marjoram et al. (2003), using a proposal density and a tolerance :

###### Algorithm 1 (Abc-Mcmc(ϵ)).

Suppose and are any starting values, such that and . For , iterate:

1. Draw and .

2. With probability

accept and set ; otherwise reject and set , where

Note that Algorithm 1 may be implemented by storing only and the related distances , and in what follows, we regard either or as the output of Algorithm 1. Note also that in practice, the initial values should be taken as the state of the Algorithm 1 run for a number of initial ‘burn-in’ iterations, during which time an adaptive algorithm for parameter tuning may be employed (Section 4).

###### Definition 2.

Suppose is the output of ABC-MCMC() for some . For any such that for some , and for any function , define

 W(ϵ0,ϵ)k :=U(ϵ0,ϵ)k∑nj=1U(ϵ0,ϵ)j, U(ϵ0,ϵ)k :=ϕ(Tk/ϵ)ϕ(Tk/ϵ0), Eϵ0,ϵ(f) :=n∑k=1W(ϵ0,ϵ)kf(Θk), Sϵ0,ϵ(f) :=n∑k=1(W(ϵ0,ϵ)k)2[f(Θk)−Eϵ0,ϵ(f)]2.

The estimator approximates and may be used to construct a confidence interval; see Algorithm 6 below. The following algorithm shows that in case of simple cut-off, and may be calculated simultaneously for all tolerances efficiently:

###### Algorithm 3.

Suppose and is the output of ABC-MCMC().

1. Sort with respect to :

• Find indices such that for all .

• Denote .

2. For all unique values , let , and define

 Eϵ0,ϵ(f):=1mϵmϵ∑k=1f(^Θk),andSϵ0,ϵ(f):=1m2ϵmϵ∑k=1[f(^Θk)−Eϵ0,ϵ(f)]2.

(and for , let and .)

The sorting in Algorithm 3(i) may be performed in time, and and may all be calculated in time by forming appropriate cumulative sums.

Theorem 5 below details consistency of , and relates

to the limiting variance, in case the following (well-known) condition ensuring a central limit theorem holds:

###### Assumption 4 (Finite integrated autocorrelation).

Suppose that and is finite, with , where is a stationary version of the ABC-MCMC() chain, and

 hϵ0,ϵ(θ,y):=wϵ0,ϵ(y)f(θ)wherewϵ0,ϵ(y):=ϕ(d(y,y∗)/ϵ)/ϕ(d(y,y∗)/ϵ0).
###### Theorem 5.

Suppose is the output of ABC-MCMC(), and denote by and the estimators in Definition 2. If is -irreducible, then, for any , we have as :

1. almost surely, whenever the expectation is finite.

2. Under Assumption 4, in distribution, where

 τϵ0,ϵ(f):=(1+2∑k≥1ρ(ϵ0,ϵ)k)∈[0,∞),andnS(n)ϵ0,ϵ(f)a.s.−−→vϵ0,ϵ(f)∈[0,∞).

Proof of Theorem 5 is given in Appendix A. Based on Theorem 5, we suggest to report the following approximate confidence intervals for the suggested estimators:

###### Algorithm 6.

Suppose is the output of ABC-MCMC() and is a function, then for any :

1. Calculate and as in Definition 2 (or in Algorithm 3).

2. Calculate , an estimate of the integrated autocorrelation of .

3. Report the confidence interval

 [Eϵ0,ϵ(f)±β√Sϵ0,ϵ(f)^τϵ0(f)],

where

corresponds to the desired normal quantile.

The classical choice for in Algorithm 6(ii) is windowed autocorrelation, , with some , where is the sample autocorrelation of (cf. Geyer, 1992), but also more sophisticated techniques for the calculation of the asymptotic variance have been suggested (e.g. Flegal and Jones, 2010).

Because computing an estimate of is computationally demanding, and because such an estimate is likely to be unstable for small , Algorithm 6 is based on the use of as a common autocorrelation for all . This relies on the approximation , which may not always be entirely accurate, but likely to be reasonable, as illustrated by Theorem 7 in Section 3 below.

We remark that, although we focus on the case of using a common cut-off for both the ABC-MCMC and post-correction, one could also consider using two different cut-offs. The extension to Definition 2 is straightforward, and Algorithm 3 holds with simple post-correction cut-off, under a support condition.

## 3. Confidence interval and efficiency

The following result, whose proof is given in Appendix A, gives an expression for the integrated autocorrelation in case of simple cut-off.

###### Theorem 7.

Suppose Assumption 4 holds and , then

 τϵ0,ϵ(f)−1=(ˇτϵ0,ϵ(f)−1)varπϵ0(fϵ0,ϵ)+2∫πϵ0(θ)¯wϵ0,ϵ(θ)(1−¯wϵ0,ϵ(θ))rϵ0(θ)1−rϵ0(θ)f2(θ)dθvarπϵ0(fϵ0,ϵ)+∫πϵ0(θ)¯wϵ0,ϵ(θ)(1−¯wϵ0,ϵ(θ))f2(θ)dθ,

where , , is the integrated autocorrelation of and the rejection probability of the ABC-MCMC() chain at .

We next discuss how this loosely suggests that . Note that , and under suitable regularity conditions both and are continuous with respect to , and as . Then, for , we have and therefore . For small , the terms with are of order , and are dominated by the other terms of order . The remaining ratio may be written as

 2∫πϵ0(θ)¯wϵ0,ϵ(θ)(1−¯wϵ0,ϵ(θ))rϵ0(θ)1−rϵ0(θ)f2(θ)dθ∫πϵ0(θ)¯wϵ0,ϵ(θ)(1−¯wϵ0,ϵ(θ))f2(θ)dθ =2Eπϵ0[¯g2ϵ0,ϵ(Θ)rϵ0(Θ)1−rϵ0(Θ)],

where with . If , then the term is upper bounded by , and we believe it to be often less than , because the latter expression is similar to the contribution of rejections to the integrated autocorrelation; see the proof of Theorem 7.

For general , it appears to be hard to obtain similar theoretical result, but we expect the approximation to be still sensible. Theorem 7 relies on being independent of conditional on , assuming at least single acceptance. This is not true with other cut-offs, but we believe that the dependence of from given is generally weaker than dependence of and , suggesting similar behaviour.

Let us state next a general upper bound for the IS-corrected ABC-MCMC as we suggest, with respect to a direct ABC-MCMC with a smaller tolerance.

###### Theorem 8.

For any , denote by the asymptotic variance of the estimator of Definition 2 (see Theorem 5(ii)). Define

 wϵ0,ϵ(y):=ϕ(d(y,y∗)/ϵ)ϕ(d(y,y∗)/ϵ0),cϵ:=∫pr(θ)Lϵ(θ)dθ, ~πϵ(θ,y):=pr(θ)g(y∣θ)Kϵ(y,y∗)cϵ,

and denote . Then for any ,

 σ2ϵ0,ϵ(f)≤cϵ0cϵ[σ2ϵ(f)+~πϵ(¯f2(1−wϵ0,ϵ))],

where is the asymptotic variance of the direct ABC-MCMC().

Theorem 8 follows directly from (Franks and Vihola, 2017, Corollary 4). The upper bound guarantees that a moderate correction, that is, close to and close to , is nearly as efficient as direct ABC-MCMC. Indeed, typically and as , in which case Theorem 8 implies . However, as , the bound becomes less informative.

## 4. A tolerance adaptive ABC-MCMC algorithm

We propose Algorithm 9 below to adapt the tolerance in the ABC-MCMC during burn-in, in order to obtain a user-specified overall acceptance rate . Together with the approach based on post-correction of Section 2, we thus obtain an automated ABC inference solution that does not require prior choice of an .

In Algorithm 9, we assume that a desired acceptance rate is specified. We used in our experiments, and discuss this choice later. We also assume a choice of decreasing positive step sizes . We used in our experiments. For convenience, we denote the distance distribution here as , where for .

###### Algorithm 9 (Ta(nb,α∗)).

Suppose is a starting value with .

1. Initialise where and .

2. For , iterate:

1. Draw .

2. Draw .

3. Accept, by setting , with probability

 (2) αϵk(Θk,Tk;Θ′k,T′k):=min{1,pr(Θ′k)q(Θ′k,Θk)ϕ(T′k/ϵk)pr(Θk)q(Θk,Θ′k)ϕ(Tk/ϵk)}

and otherwise reject, by setting .

3. Output .

The following simple conditions suffice for convergence of the adaptation

###### Assumption 10.

Suppose and the following hold:

1. with and a constant.

2. The domain , , is a nonempty open set.

3. is uniformly bounded on , and for .

4. is a uniformly bounded density, and on , .

5. admits a uniformly bounded density .

6. stays in a set almost surely, where .

7. for all .

###### Theorem 11.

Under Assumption 10, the expected value of the acceptance probability (2), taken with respect to the stationary measure of the chain, converges to .

Proof of Theorem 11 will follow from the more general Theorem 13 of Appendix B. Theorem 13 is phrased for geometrically ergodic chains on possibly unbounded domains without the lower bound in Assumption 10(iv). See Appendix C for the proofs of both theorems.

In practice, the tolerance adaptation is the most straightforward to apply with a symmetric random walk proposal adapted simultaneously with proposal covariance adaptation (Andrieu and Moulines, 2006; Haario et al., 2001) (see Algorithm 22 of Appendix D for a detailed description of the resulting algorithm). Such simultaneous use of different optimisation criteria within adaptive MCMC has been discussed, for example, in the review Andrieu and Thoms (2008). While we do not consider Algorithm 22 explicitly in our theoretical analysis, our results could be elaborated, along the lines of Andrieu and Moulines (2006), to accommodate Algorithm 22 in detail.

In the standard Adaptive Metropolis algorithm (Haario et al., 2001), the limiting acceptance rate is often around (Roberts et al., 1997). In the ABC-MCMC context, this acceptance rate would be reached if the tolerance would be made infinite, and if the prior distribution would be regular enough (e.g. Gaussian). Because the mean acceptance rate of ABC-MCMC typically decreases when tolerance is decreased (see Lemma 15 of Appendix C in case of ), and because the likelihood approximation must be reasonable, the desired acceptance rate should be substantially lower than .

As ABC-MCMC may be interpreted as an instance of pseudo-marginal MCMC, for which there are certain conditions under which the optimal acceptance rate of about is reached (Sherlock et al., 2015), one could take this as a guideline as well for the ABC-MCMC. However, the context of Sherlock et al. (2015) is quite dissimilar to that of ABC-MCMC, and so we decided to push the acceptance rate a little higher to ensure sufficient mixing. As well, we do subsequent post-correction, which is further justification for a slightly inflated tolerance and therefore acceptance rate.

## 5. Experiments

We experiment with our methods on two models, a lightweight Gaussian toy example, and a Lotka-Volterra model. Our experiments aim at providing information regarding the following questions:

• Can ABC-MCMC with larger tolerance and post-correction to a desired tolerance deliver more accurate results than direct application of ABC-MCMC?

• Does our approximate confidence interval appear reliable?

• How well does the adaptive ABC-MCMC work in practice?

In all our experiments, we apply the Adaptive Metropolis (Haario et al., 2001; Andrieu and Moulines, 2006) covariance adaptation, which is run during the whole simulation, using an identity covariance initially.

Regarding our first question, we investigate running the ABC-MCMC starting near the posterior mode with different pre-selected tolerances, both selected in a preliminary pilot experiment. We first attempted to perform the experiments by initialising the chains from independent samples of the prior distribution, but in this case, most of the chains did not accept a single move during the whole run. In contrast, our experiments with tolerance adaptation do start from initial points drawn from the prior distribution, and both the tolerances and the covariances are adjusted fully automatically by our algorithm. The latter assumes no prior information of the model at all, which we aim at.

In our tests about confidence intervals, we employ a simple ‘automatic window’ estimator of integrated autocorrelation of the form , where are lag- sample autocorrelations, and where is the smallest positive integer such that (Sokal, 1996).

When running the covariance adaptation alone, we employ the covariance adaptation of Andrieu and Moulines (2006) with step size , which behaves similar to the original Adaptive Metropolis algorithm of Haario et al. (2001). In case we apply tolerance adaptation, we use step size for both the tolerance adaptation and for the covariance adaptation. Slower decaying step sizes such as this often behave better with acceptance rate adaptation (cf. Vihola, 2012, Remark 3).

All the experiments are implemented in Julia (Bezanson et al., 2017), and the codes are available in https://bitbucket.org/mvihola/abc-mcmc.

### 5.1. One-dimensional Gaussian model

Our first model is a toy model with and , where

is standard Gaussian random variable. The true posterior without ABC approximation is Gaussian. While this scenario is clearly academic, the prior is far from the posterior, which we believe to be common in practice. It is clear that

has zero mean for all , and also that the distribution is spread wider for bigger . We experiment with both simple cut-off and Gaussian cut-off .

We run the experiments with 10,000 independent chains, each for 11,000 iterations including 1,000 burn-in. The chains were always started from . Figures 1 and 2 show results of the same experiments with simple and Gaussian cut-off. On the left, a single realisation of the estimates and confidence intervals calculated for all are shown for functions (above) and (below). The figures on the right show box plots of the final estimators calculated for each chain, for five equispaced tolerance values between and ( axis labels indicate these tolerances). The leftmost box plot in each group corresponds to the direct ABC-MCMC targeting that tolerance, and the rightmost box plot corresponds to the post-corrected estimators from the ABC-MCMC with , and the second from the right with etc. The colour indicates the . Some post-corrected estimates appear to be slightly more accurate than ABC-MCMC, and in the results suggest that might be a good choice, if the desired tolerance is .

Table 1 indicates the frequencies of the calculated 95% confidence intervals containing the ‘ground truth’, over the 10,000 independent experiments, as well as acceptance rates. The ground truth for is known to be zero for all , and the overall mean of all the calculated estimates is used as the ground truth for . The frequencies appear close to ideal with the post-correction approach, being slightly pessimistic in case of simple cut-off as anticipated by the theoretical considerations (see Theorem 7 and discussion below).

Figure 3 shows progress of tolerance adaptations during the burn-in, and histogram of the mean acceptance rates of the chain after burn-in. The lines on the left show the median, and the shaded regions indicate the 50%, 75%, 95% and 99% quantiles. The figures indicate concentration, but suggest that the adaptation has not fully converged yet. This is also indicated by the mean acceptance rate over all realisations, which are and with simple and Gaussian cutoff, respectively. Table 2 shows root mean square errors from the ground truth, for both the fixed tolerance estimators, and the adaptive algorithms, for tolerance . Here, only the adaptive chains with final tolerance were included (9,997 and 9,996 out of 10,000 chains for the simple and Gaussian cut-offs, respectively).

### 5.2. Lotka-Volterra model

Our second experiment is a Lotka-Volterra model suggested in (Boys et al., 2008), and also analysed in the ABC context in (Fearnhead and Prangle, 2012). The model is a Markov process of counts, corresponding to a reaction network with rate , with rate and with rate . The reaction rates are parameters, which we equip here with a uniform prior, . The data is a simulated trajectory from the model with until time . The ABC is based on Euclidean distance of a six-dimensional summary statistic, which consists of:

• Sample autocorrelation of at lag 10, multiplied by 100.

• 10% and 90% (time-averaged) quantiles of both and .

• Number of jumps (or events), divided by 10.

The summary statistics are then for the observed series.

We first run comparisons similar to Section 5.1, but now only with 1,000 independent chains and simple cut-off. We investigate the effect of post-correction, with 20,000 samples, including 10,000 burn-in, for each chain. The MCMC is run on log-transformed , and all chains were started from near the mode, from . Figure 4 and Table 3 show similar comparisons as in Section 5.1. The results suggest that post-corrected ABC does provide slightly more accurate estimators, particularly with smaller tolerances.

In addition, we experiment with the tolerance adaptation, using also 20,000 samples out of which 10,000 are burn-in. Figure 5 shows the progress of the -tolerance during the burn-in, and histogram of the realised mean acceptance rates during the estimation phase. The realised acceptance rates are concentrated around the desired , with mean . Table 4 shows RMSEs of both the fixed tolerance ABC-MCMC outputs and with tolerance adaptation. Again, only the adaptive chains with final tolerance were included (999 out of 1,000 chains).

In this case, the chains run with the tolerance adaptation led to better results than those run only with the covariance adaptation (and fixed tolerance). This perhaps surprising result may be due to the initial behaviour of the covariance adaptation, which may be unstable when there are many rejections. Different initialisation strategies, for instance following (Haario et al., 2001, Remark 2), might lead to more stable behaviour compared to using the adaptation of Andrieu and Moulines (2006) from the start, as we do. The different step size sequences ( and ) could also play a rôle. We repeated the experiment for the chains with fixed tolerances, but now with covariance adaptation step size . This led to more stable behaviour of the ABC-MCMC with tolerance . In any case, also here, the adaptive ABC-MCMC using the tolerance adaptation delivered slightly better results (see supplementary results in Appendix E).

## 6. Discussion

We believe our approach consisting of ABC-MCMC with post-processing is a useful addition, and complements some earlier and related work. As previously mentioned, trimming of ABC-MCMC output to finer tolerances has been considered earlier (e.g. Wegmann et al., 2009). Our experimental results suggest that this can indeed be beneficial, and our confidence interval may make the approach more appealing in practice.

Another related approach by Bortot et al. (2007) makes tolerance an auxiliary variable with a user-specified prior, and ABC-MCMC is run targeting the joint posterior of parameter and tolerance. While this approach avoids tolerance selection, we believe that our approach, where the effect of tolerance can be investigated explicitly, can be helpful in interpretation of the ABC posterior. In fact, Bortot et al. (2007) also provide tolerance-dependent analysis, but we believe that our estimators, with associated confidence intervals, have a more immediate interpretation.

Automatic selection of tolerance in ABC-MCMC has been considered earlier in Ratmann et al. (2007), who propose an algorithm based on tempering and a cooling schedule. It has been remarked by Sisson and Fan (2018) that acceptance rate based adaptation could be used to deal with the choice of a suitable tolerance. Based on our experiments, the adaptive ABC-MCMC we present in this paper appears to perform well in practice, and provides reliable results with post-correction. The tolerance adaptation also seems to benefit the covariance adaptation in the early phases. For the adaptive ABC-MCMC to work efficiently, the MCMC chains must be taken relatively long, rendering the approach difficult for computationally demanding models. However, we believe that our approach using adaptive ABC-MCMC provides a straightforward way to do inference with ABC models.

Our estimators, and their uncertainty estimators, could also turn out to be useful in the regression adjustment context (Wegmann et al., 2009; Beaumont et al., 2002; Blum, 2010). We did not consider such adjustments, but note that approximate normality and the confidence bounds may be used to derive an appropriately weighted estimator that reflects the uncertainty of the estimators.

We conclude with a brief discussion of certain extensions of the suggested post-correction method. The first extension is based on ‘recycling’ the rejected samples in the estimator (Ceperley et al., 1977). This may improve the accuracy (but can also reduce accuracy in certain pathological cases; see Delmas and Jourdain (2009)). The ‘waste recycling’ estimator is

 EWRϵ0,ϵ(f):=n∑k=1W(ϵ0,ϵ)k[αϵ0(Θk,Yk;~Θk+1,~Yk+1)f(~Θk+1)+[1−αϵ0(Θk,Yk;~Θk+1,~Yk+1)]f(Θk)].

When is consistent under Theorem 5(i), this is also a consistent estimator. Namely, as in the proof (in Appendix A) of Theorem 5, we find that is a Harris recurrent Markov chain with invariant distribution

 ^πϵ0(θ,y,~θ,~y)=~πϵ0(θ,y)~q(θ,y;~θ,~y),

and , where . Therefore, is a strongly consistent estimator of

 E^πϵ[αϵ0(Θ,Y;~Θ,~Y)f(~Θ)+[1−αϵ0(Θ,Y;~Θ,~Y)]f(Θ)]=Eπϵ[f(Θ)].

See (Rudolf and Sprungk, 2018; Schuster and Klebanov, 2018) for alternative waste recycling estimators based on importance sampling analogues.

Another extension, which could be considered, is about enhancing the accuracy of the estimator with smaller values of , by performing further simulations from the model (which may be calculated in parallel for different ). Namely, a new estimator may be formed as follows:

 ^Eϵ0,ϵ(f)=∑nk=1∑mj=0^U(ϵ0,ϵ)k,jf(Θk)∑nℓ=1∑mi=0^U(ϵ0,ϵ)ℓ,i,^U(ϵ0,ϵ)k,0:=U(ϵ0,ϵ)kand% ^U(ϵ0,ϵ)k,j:=^Nkϕ(^Tk,j/ϵ)ϕ(Tk/ϵ0),

for , where is the number of independent random variables generated before observing where , and with independent . This ensures that

 E[^Nkϕ(^Tk,j/ϵ)∣Θk=θ,Yk=y]=Lϵ(θ)Pg(⋅∣θ)(ϕ(d(Y,y∗)/ϵ0)>0),

which is sufficient to ensure that is a proper weighting scheme from to ; see (Vihola et al., 2016, Proposition 17(ii)), and consequently the average is a proper weighting.

The latter extension, which involves additional simulations as post-processing, is similar to ‘lazy ABC’, which incorporates a randomised stopping rule for simulation (Prangle, 2016, 2015), and to unbiased ‘exact’ ABC (Tran and Kohn, 2015), which may lead to estimators which get rid of -bias entirely, using the debiasing approach lately investigated in (Rhee and Glynn, 2015; McLeish, 2011).

## 7. Acknowledgements

This work was supported by Academy of Finland (grants 274740, 284513 and 312605). The authors wish to acknowledge CSC, IT Center for Science, Finland, for computational resources. The authors wish to thank Christophe Andrieu for useful discussions.

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