Published on May 6, 2026
Markov chain Monte Carlo (MCMC) methods have long been a staple in statistical analysis, offering a means of sampling from complex distributions. Traditionally, effective sample size has been a key metric associated with these methods. However, analysts often rely on scalar or Euclidean summaries, leading to issues when applied to manifold-valued samples.
Recent developments propose an intrinsic effective sample size based on kernel discrepancy. This method eliminates ambiguity that remain consistent despite transformations like rotations or changes in coordinate charts. The new approach captivates researchers with its ability to yield an accurate measure of independent draws needed for comparing empirical distributions against target distributions.
Through rigorous analysis, this innovative framework offers an exact finite-sample risk interpretation and shows asymptotic relationships that enhance its practical application. The proposed methodology also establishes kernel invariance and provides insights into effective kernel constructions, addressing limitations found in conventional techniques. Sphere experiments demonstrate the merits of rotation invariance and the calibration of results against empirical errors.
The implications of this research are significant. evaluation of effective sample sizes, researchers can achieve more reliable estimates in various fields that utilize MCMC methods. This advancement promises to enhance the accuracy and applicability of statistical modeling on complex geometrical spaces, potentially transforming how researchers approach manifold-valued data.
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