Unified Unbiased Variance Estimation for MMD: Robust Finite-Sample Performance with Imbalanced Data and Exact Acceleration under Null and Alternative Hypotheses

The maximum mean discrepancy (MMD) is a kernel-based nonparametric statistic for two-sample testing, whose inferential accuracy depends critically on variance characterization. Existing work provides various finite-sample estimators of the MMD variance, often differing under the null and alternative hypotheses and across balanced or imbalanced sampling schemes. In this paper, we study the variance of the MMD statistic through its U-statistic representation and Hoeffding decomposition, and establish a unified finite-sample characterization covering different hypotheses and sample configurations. Building on this analysis, we propose an exact acceleration method for the univariate case under the Laplacian kernel, which reduces the overall computational complexity from $\mathcal O(n^2)$ to $\mathcal O(n \log n)$.

Goal-Oriented Bayesian Optimal Experimental Design for Nonlinear Models using Markov Chain Monte Carlo

Optimal experimental design (OED) provides a systematic approach to quantify and maximize the value of experimental data. Under a Bayesian approach, conventional OED maximizes the expected information gain (EIG) on model parameters. However, we are often interested in not the parameters themselves, but predictive quantities of interest (QoIs) that depend on the parameters in a nonlinear manner. We present a computational framework of predictive goal-oriented OED (GO-OED) suitable for nonlinear observation and prediction models, which seeks the experimental design providing the greatest EIG on the QoIs. In particular, we propose a nested Monte Carlo estimator for the QoI EIG, featuring Markov chain Monte Carlo for posterior sampling and kernel density estimation for evaluating the posterior-predictive density and its Kullback-Leibler divergence from the prior-predictive. The GO-OED design is then found by maximizing the EIG over the design space using Bayesian optimization. We demonstrate the effectiveness of the overall nonlinear GO-OED method, and illustrate its differences versus conventional non-GO-OED, through various test problems and an application of sensor placement for source inversion in a convection-diffusion field.