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

2026-05-04

AI Summary

Traditional effective sample size (ESS) estimators rely on Euclidean coordinates and fail to preserve geometric invariance in manifold-valued MCMC settings. This work proposes the first intrinsic, coordinate-free definition of ESS, grounded in the squared kernel discrepancy in a reproducing kernel Hilbert space to quantify the deviation between the empirical and target distributions. The approach integrates manifold geometry with translation-invariant kernels, offering both a finite-sample risk interpretation and an asymptotic autocorrelation expression, while guaranteeing geometric consistency when the kernel and manifold are compatible. The estimator is shown to be consistent under boundedness and absolute regularity conditions, and experiments on the sphere demonstrate its rotational invariance and well-calibrated sensitivity to distributional errors.

Abstract

Effective sample size is a standard summary of Markov chain Monte Carlo output, but it is usually attached to scalar or Euclidean summaries chosen by the analyst. For manifold-valued samples this choice is not canonical: coordinate-wise effective sample sizes can change under rotations, chart changes, or alternative embeddings of the same underlying path. We propose an intrinsic effective sample size based on kernel discrepancy. The proposed quantity is the number of independent draws that would yield the same expected squared kernel discrepancy between the empirical distribution and the target distribution. This gives an exact finite-sample risk interpretation, an asymptotic integrated-autocorrelation representation, and a coordinate-free diagnostic whenever the kernel respects the geometry of the state space. We establish invariance under transported kernels, operator and principal-direction interpretations, and consistency of a lag-window estimator under boundedness and absolute-regularity conditions. We also discuss valid kernel constructions on manifolds, emphasizing that geodesic Gaussian kernels are not generally positive definite on curved spaces. Sphere experiments illustrate rotation invariance and calibration of the proposed diagnostic against empirical distributional error.

Problem

• effective sample size
• manifold-valued data
• Markov chain Monte Carlo
• kernel discrepancy
• coordinate invariance

Innovation

• intrinsic effective sample size
• manifold-valued MCMC
• kernel discrepancy
• geometric invariance
• positive definite kernels on manifolds