Optimally-Weighted Estimators of the Maximum Mean Discrepancy for Likelihood-Free Inference

TL;DR

This paper introduces a new estimator for the maximum mean discrepancy (MMD) that reduces the number of samples needed for accurate likelihood-free inference, especially useful for costly simulations.

Contribution

The authors propose an optimally-weighted MMD estimator with improved sample complexity, enhancing efficiency in likelihood-free inference for expensive simulators.

Findings

The new estimator achieves lower sample complexity than traditional methods.

Theoretical analysis confirms improved convergence properties.

Simulation studies demonstrate practical effectiveness on benchmark problems.

Abstract

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.