Copula Approximate Bayesian Computation Using Distribution Random Forests

This paper 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 likelihood functions, which unifies ad extends previous ABC methods. This framework can 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 account for skewness and accurately estimated by Distribution Random Forests (\texttt{drf}) while performing automatic summary statistics (covariates) selection, and on robustly-estimated copula dependence parameters. Also, the framework provides a novel multivariate mode estimator to perform for MLE and posterior mode estimation, and an optional step to perform model selection from a given set of models with posterior probabilities estimated by \texttt{drf}. The posterior distribution estimation accuracy of the ABC framework is illustrated and compared with previous standard ABC methods, through simulation studies involving low- and high-dimensional models with computable posterior distributions which are either unimodal, skewed, and multimodal; and exponential random graph and mechanistic network models which are each defined by an intractable likelihood from which it is costly to simulate large network datasets. This paper also introduces and studies a new new solution to simulation cost problem in ABC. Finally, the new framework is illustrated through analyses of large real-life networks of sizes ranging between 28,000 to 65.6 million nodes (between 3 million to 1.8 billion edges), including a large multilayer network with weighted directed edges. Keywords: Bayesian analysis, Maximum Likelihood, Intractable likelihood.