Approximate Bayesian Computation (ABC) methods are applicable to statistical models specified by generative processes with analytically intractable likelihoods. These methods try to approximate the posterior density of a model parameter by comparing the observed data with additional process-generated simulated datasets. For computational benefit, only the values of certain well-chosen summary statistics are usually compared, instead of the whole dataset. Most ABC procedures are computationally expensive, justified only heuristically, and have poor asymptotic properties. In this article, we introduce a new empirical likelihood-based approach to the ABC paradigm called ABCel. The proposed procedure is computationally tractable and approximates the target log posterior of the parameter as a sum of two functions of the data -- namely, the mean of the optimal log-empirical likelihood weights and the estimated differential entropy of the summary functions. We rigorously justify the procedure via direct and reverse information projections onto appropriate classes of probability densities. Past applications of empirical likelihood in ABC demanded constraints based on analytically tractable estimating functions that involve both the data and the parameter; although by the nature of the ABC problem such functions may not be available in general. In contrast, we use constraints that are functions of the summary statistics only. Equally importantly, we show that our construction directly connects to the reverse information projection. We show that ABCel is posterior consistent and has highly favourable asymptotic properties. Its construction justifies the use of simple summary statistics like moments, quantiles, etc, which in practice produce an accurate approximation of the posterior density. We illustrate the performance of the proposed procedure in a range of applications.

翻译：近似贝叶斯计算（ABC）方法适用于由生成过程定义的统计模型，其似然函数无法解析求解。该方法通过将观测数据与额外生成的模拟数据集进行比较，来近似模型参数的后验密度。为降低计算成本，通常仅比较精心选择的汇总统计量而非完整数据集。大多数ABC程序计算代价高昂，仅凭启发式验证，且渐近性质较差。本文提出一种新的基于经验似然的ABC范式方法，命名为ABCel。该程序在计算上易于处理，将目标参数的对数后验近似为两个数据函数的和——即最优对数经验似然权重的均值与汇总函数估计微分熵。我们通过直接和逆向信息投影到适当的概率密度族上严格证明了该方法的合理性。过去经验似然在ABC中的应用依赖于包含数据和参数的解析可计算估计函数约束；然而由于ABC问题的本质，此类函数通常不可得。与此相反，我们采用仅依赖于汇总统计量的函数作为约束。同等重要的是，我们证明该构造直接关联至逆向信息投影。我们证明ABCel具有后验一致性和高度优良的渐近性质。其构造使得矩、分位数等简单汇总统计量的使用得到合理解释，这些统计量在实践中能精确近似后验密度。我们通过一系列应用实例展示了所提方法的性能。