We consider the asymptotic properties of Approximate Bayesian Computation (ABC) for the realistic case of summary statistics with heterogeneous rates of convergence. We allow some statistics to converge faster than the ABC tolerance, other statistics to converge slower, and cover the case where some statistics do not converge at all. We give conditions for the ABC posterior to converge, and provide an explicit representation of the shape of the ABC posterior distribution in our general setting; in particular, we show how the shape of the posterior depends on the number of slow statistics. We then quantify the gain brought by the local linear post-processing step.

翻译：我们考虑在汇总统计量具有异质收敛速率的实际情况下，近似贝叶斯计算（ABC）的渐近性质。我们允许部分统计量比ABC容差收敛更快，部分统计量收敛更慢，并涵盖某些统计量完全不收敛的情形。我们给出了ABC后验分布的收敛条件，并提供了在一般设定下ABC后验分布形状的显式表示；特别地，我们揭示了后验形状如何依赖于慢速统计量的数量。随后，我们量化了局部线性后处理步骤带来的增益。