Ongoing modern computational advancements continue to make it easier to collect increasingly large and complex datasets, which can often only be realistically analyzed using models defined by intractable likelihood functions. This Stats invited feature article introduces and provides an extensive simulation study of a new approximate Bayesian computation (ABC) framework for estimating the posterior distribution and the maximum likelihood estimate (MLE) of the parameters of models defined by intractable likelihoods, that unifies and extends previous ABC methods proposed separately. This framework, copulaABCdrf, aims to accurately estimate and describe the possibly skewed and high-dimensional posterior distribution by a novel multivariate copula-based meta-t distribution based on univariate marginal posterior distributions that can be accurately estimated by distribution random forests (drf), while performing automatic summary statistics (covariates) selection, based on robustly estimated copula dependence parameters. The copulaABCdrf framework also provides a novel multivariate mode estimator to perform MLE and posterior mode estimation and an optional step to perform model selection from a given set of models using posterior probabilities estimated by drf. The posterior distribution estimation accuracy of the ABC framework is illustrated and compared with previous standard ABC methods through several simulation studies involving low- and high-dimensional models with computable posterior distributions, which are either unimodal, skewed, or multimodal; and exponential random graph and mechanistic network models, each defined by an intractable likelihood from which it is costly to simulate large network datasets. This paper also proposes and studies a new solution to the simulation cost problem in ABC involving the posterior estimation of parameters from datasets simulated from the given model that are smaller compared to the potentially large size of the dataset being analyzed. This proposal is motivated by the fact that, for many models defined by intractable likelihoods, such as the network models when they are applied to analyze massive networks, the repeated simulation of large datasets (networks) for posterior-based parameter estimation can be too computationally costly and vastly slow down or prohibit the use of standard ABC methods. The copulaABCdrf framework and standard ABC methods are further illustrated through analyses of large real-life networks of sizes ranging between 28,000 and 65.6 million nodes (between 3 million and 1.8 billion edges), including a large multilayer network with weighted directed edges. The results of the simulation studies show that, in settings where the true posterior distribution is not highly multimodal, copulaABCdrf usually produced similar point estimates from the posterior distribution for low-dimensional parametric models as previous ABC methods, but the copula-based method can produce more accurate estimates from the posterior distribution for high-dimensional models, and, in both dimensionality cases, usually produced more accurate estimates of univariate marginal posterior distributions of parameters. Also, posterior estimation accuracy was usually improved when pre-selecting the important summary statistics using drf compared to ABC employing no pre-selection of the subset of important summaries. For all ABC methods studied, accurate estimation of a highly multimodal posterior distribution was challenging. In light of the results of all the simulation studies, this article concludes by discussing how the copulaABCdrf framework can be improved for future research.
