Copula Approximate Bayesian Computation Using Distribution Random Forests

This 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, which unifies and extends previous ABC method. This framework, copulaABcdrf, aims to accurately estimate and describe the possibly skewed and high dimensional posterior distribution by a novel multivariate copula-based meta-\textit{t} distribution, based on univariate marginal posterior distributions which can be accurately estimated by Distribution Random Forests (drf), while performing automatic summary statistics (covariates) selection, and robust estimation of 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 copulaABcdrf is illustrated and compared to 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. We also study a new solution to the simulation cost problem in ABC. The copulaABcdrf framework and standard ABC methods are further illustrated through analyses of large real-life networks. The results of the simulation and empirical studies, and their implications for future research, are summarized. Keywords: Bayesian analysis, Maximum Likelihood, Intractable likelihood.