Surrogate models can be used to accelerate approximate Bayesian computation (ABC). In one such framework the discrepancy between simulated and observed data is modelled with a Gaussian process. So far principled strategies have been proposed only for sequential selection of the simulation locations. To address this limitation, we develop Bayesian optimal design strategies to parallellise the expensive simulations. We also touch the problem of quantifying the uncertainty of the ABC posterior due to the limited budget of simulations. Approximate Bayesian computation (Marin et al., 2012; Lintusaari et al., 2017 ) is used for Bayesian inference when the analytic form of the likelihood function of a statistical model of interest is either unavailable or too costly to evaluate, but simulating the model is feasible. Unfortunately, many models e.g. in genomics and epidemiology (Numminen et al., 2013; Marttinen et al., 2015; McKinley et al., 2018) and climate science (Holden et al., 2018) are costly to simulate making sampling-based ABC inference algorithms infeasible. To increase sample-efficiency of ABC, various methods using surrogate models such as neural networks (Papamakarios and Murray, 2016; Papamakarios et al., 2019; Lueckmann et al., 2019; Greenberg et al., 2019) and Gaussian processes (Meeds and Welling, 2014; Wilkinson, 2014; Gutmann and Corander, 2016; Järvenpää et al., 2018 Järvenpää et al., , 2019a have been proposed. In one promising surrogate-based ABC framework the discrepancy between the observed and simulated data is modelled with a Gaussian process (GP) (Gutmann and Corander, 2016; Järvenpää et al., 2018 Järvenpää et al., , 2019a . Sequential Bayesian experimental design (or active learning) methods to select the simulation locations so as to maximise the sample-efficiency in this framework were proposed by Järvenpää et al. (2019a) . However, one often has access to multiple computers to run some of the simulations in parallel. In this work, motivated by the related problem of batch Bayesian optimisation (Ginsbourger et al., 2010; Desautels et al., 2014; Shah and Ghahramani, 2015; Wu and Frazier, 2016) and the parallel GP-based method by Järvenpää et al. (2019b) for inference tasks where noisy and potentially expensive log-likelihood evaluations can be obtained, we resolve this limitation by developing principled batch simulation methods which considerably decrease the wall-time needed for ABC inference. The posterior distribution is often summarised for further decision making using e.g. expectation and variance. When the computational resources for ABC inference are limited, it would be important to assess the accuracy of such summaries, but this has not been explicitly acknowledged in earlier work. We devise an approximate numerical method to propagate the uncertainty of the discrepancy, represented by the GP model, to the resulting ABC posterior summaries. We call our resulting framework as Bayesian ABC in analogy with the related problems of Bayesian quadrature (O'Hagan, 1991; Osborne et al., 2012; Briol et al., 2019) and Bayesian optimisation (BO) (Brochu et al., 2010; Shahriari et al., 2015) .
We propose principled batch Bayesian experimental design strategies and a method for uncertainty quantification of the posterior summaries in a Gaussian process surrogate-based approximate Bayesian computation framework.
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