Likelihood-free inference methods typically make use of a distance between simulated and real data. A common example is the maximum mean discrepancy (MMD), which has previously been used for approximate Bayesian computation, minimum distance estimation, generalised Bayesian inference, and within the nonparametric learning framework. The MMD is commonly estimated at a root-$m$ rate, where $m$ is the number of simulated samples. This can lead to significant computational challenges since a large $m$ is required to obtain an accurate estimate, which is crucial for parameter estimation. In this paper, we propose a novel estimator for the MMD with significantly improved sample complexity. The estimator is particularly well suited for computationally expensive smooth simulators with low- to mid-dimensional inputs. This claim is supported through both theoretical results and an extensive simulation study on benchmark simulators.

翻译：无似然推断方法通常利用模拟数据与真实数据之间的距离。一个常见的例子是最大均值差异（MMD），该方法此前已被用于近似贝叶斯计算、最小距离估计、广义贝叶斯推断以及非参数学习框架中。MMD通常以根号m的速率进行估计，其中m为模拟样本数量。由于需要大量样本才能获得精确估计，这会导致显著的计算挑战——而精确估计对参数估计至关重要。本文提出了一种新的MMD估计器，该估计器显著提升了样本复杂度。该估计器特别适用于输入维度中等偏低且计算成本高昂的平滑模拟器。这一结论同时得到了理论结果与基准模拟器上大规模仿真研究的支持。