Measuring Sample Quality with Copula Discrepancies

• URL: http://arxiv.org/abs/2507.21434v1
• Date: Tue, 29 Jul 2025 02:11:45 GMT
• Title: Measuring Sample Quality with Copula Discrepancies
• Authors: Agnideep Aich, Ashit Baran Aich, Bruce Wade,
• Abstract summary: Copula Discrepancy (CD) is a principled and computationally efficient diagnostic for dependence structure.<n>Our theoretical framework provides the first structure-aware diagnostic specifically designed for the era of approximate inference.<n>With computational overhead orders of magnitude lower than existing Stein discrepancies, the CD provides both immediate practical value for MCMC practitioners and a theoretical foundation for the next generation of structure-aware sample quality assessment.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The scalable Markov chain Monte Carlo (MCMC) algorithms that underpin modern Bayesian machine learning, such as Stochastic Gradient Langevin Dynamics (SGLD), sacrifice asymptotic exactness for computational speed, creating a critical diagnostic gap: traditional sample quality measures fail catastrophically when applied to biased samplers. While powerful Stein-based diagnostics can detect distributional mismatches, they provide no direct assessment of dependence structure, often the primary inferential target in multivariate problems. We introduce the Copula Discrepancy (CD), a principled and computationally efficient diagnostic that leverages Sklar's theorem to isolate and quantify the fidelity of a sample's dependence structure independent of its marginals. Our theoretical framework provides the first structure-aware diagnostic specifically designed for the era of approximate inference. Empirically, we demonstrate that a moment-based CD dramatically outperforms standard diagnostics like effective sample size for hyperparameter selection in biased MCMC, correctly identifying optimal configurations where traditional methods fail. Furthermore, our robust MLE-based variant can detect subtle but critical mismatches in tail dependence that remain invisible to rank correlation-based approaches, distinguishing between samples with identical Kendall's tau but fundamentally different extreme-event behavior. With computational overhead orders of magnitude lower than existing Stein discrepancies, the CD provides both immediate practical value for MCMC practitioners and a theoretical foundation for the next generation of structure-aware sample quality assessment.

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