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。一个常见例子是最大均值差异（MMD），它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$m$是模拟样本的数量。这可能导致显著的计算挑战，因为需要较大的$m$才能获得精确估计，而这对参数估计至关重要。本文提出了一种新颖的MMD估计量，其样本复杂度显著改善。该估计量特别适用于计算成本高昂、输入维度中低的平滑模拟器。该论断得到了理论结果以及在基准模拟器上广泛模拟研究的支持。