Intrinsic effective sample size for manifold-valued Markov chain Monte Carlo via kernel discrepancy

In his recent paper, Kisung You proposes an innovative approach to evaluate effective sample size in manifold-valued Markov chain Monte Carlo (MCMC) using kernel discrepancy. Traditional methods often rely on scalar summaries, which can be inconsistent across different coordinate systems. By introducing an intrinsic effective sample size, You provides a framework that maintains accuracy regardless of the coordinate representation, allowing for a more reliable analysis of MCMC outputs. This method offers an exact finite-sample risk interpretation and is invariant under transformations, making it particularly useful for complex geometrical spaces.

The paper also highlights valid kernel constructions on manifolds, addressing the limitations of geodesic Gaussian kernels in curved spaces. Through experiments involving spherical geometries, You demonstrates the robustness of the proposed diagnostic in maintaining rotation invariance and calibrating against empirical distributional errors. This advancement not only allows statisticians and data scientists to gain clearer insights into their MCMC analyses but also enhances the overall reliability of sampling methods in high-dimensional and non-Euclidean settings.