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Copula Discrepancy: Benchmarking Dependence Structure

Agnideep Aich, Ashit Baran Aich

TL;DR

This paper introduces Copula Discrepancy (CD), a lightweight, dependence-focused statistic for benchmarking how well a sample preserves a fixed target copula, independent of marginals. It defines two estimation paths— a fast moment-based estimator and a more powerful MLE-based estimator—along with information-theoretic extensions CKL and CED to quantify differences in the copula density via Kendall’s tau and entropy-based measures. The authors establish consistency, asymptotic normality (for the moment-based CD), and robustness properties, and demonstrate that CD can reliably distinguish on-target from off-target dependence even when Kendall’s tau is matched, including under tail-structure mismatches. Through three controlled experiments, CD proves useful for dependence-aware tuning of biased samplers (e.g., SGLD), diagnosing tail dependence, and providing a complementary perspective to KSD. The framework scales to practical settings with millisecond-level overhead and offers a principled, interpretable supplement to omnibus discrepancy measures for dependence analysis in Bayesian computation and generative modeling.

Abstract

We study a simple statistic for benchmarking how well a sample preserves a known bivariate dependence structure. Given a target copula family (Clayton or Gumbel) and parameter $θ_P$, the Copula Discrepancy (CD) compares the target Kendall's tau $τ(θ_P)$ with the Kendall's tau implied by a parameter $\hatθ$ fitted to the sample within the target family, i.e., $|τ(θ_P)-τ(\hatθ)|$. We develop a moment-based version, prove consistency, asymptotic normality, and robustness results under i.i.d.\ sampling, and use an MLE-based version empirically for greater power against tail-structure misspecification. Building on this, we define two information-theoretic copula summaries, a copula KL divergence (CKL) and a copula entropy gap (CED), and establish basic consistency and central limit results for their plug-in estimators. In controlled experiments, CD reliably separates on-target and off-target copulas with matched Kendall's $τ$, provides a dependence-aware signal for tuning SGLD step sizes where Effective Sample Size favors overly aggressive (and biased) settings, and remains stably nonzero under deliberate tail-dependence mismatch where a naive $τ$-based diagnostic fails; CKL and CED offer a complementary Shannon-style view that echoes these findings. Timing benchmarks show that both CD variants incur only millisecond-level overhead over the tested range and exhibit near-linear empirical scaling in sample size, providing a lightweight, dependence-focused complement to quadratic-cost omnibus discrepancies such as the Kernel Stein Discrepancy (KSD).

Copula Discrepancy: Benchmarking Dependence Structure

TL;DR

This paper introduces Copula Discrepancy (CD), a lightweight, dependence-focused statistic for benchmarking how well a sample preserves a fixed target copula, independent of marginals. It defines two estimation paths— a fast moment-based estimator and a more powerful MLE-based estimator—along with information-theoretic extensions CKL and CED to quantify differences in the copula density via Kendall’s tau and entropy-based measures. The authors establish consistency, asymptotic normality (for the moment-based CD), and robustness properties, and demonstrate that CD can reliably distinguish on-target from off-target dependence even when Kendall’s tau is matched, including under tail-structure mismatches. Through three controlled experiments, CD proves useful for dependence-aware tuning of biased samplers (e.g., SGLD), diagnosing tail dependence, and providing a complementary perspective to KSD. The framework scales to practical settings with millisecond-level overhead and offers a principled, interpretable supplement to omnibus discrepancy measures for dependence analysis in Bayesian computation and generative modeling.

Abstract

We study a simple statistic for benchmarking how well a sample preserves a known bivariate dependence structure. Given a target copula family (Clayton or Gumbel) and parameter , the Copula Discrepancy (CD) compares the target Kendall's tau with the Kendall's tau implied by a parameter fitted to the sample within the target family, i.e., . We develop a moment-based version, prove consistency, asymptotic normality, and robustness results under i.i.d.\ sampling, and use an MLE-based version empirically for greater power against tail-structure misspecification. Building on this, we define two information-theoretic copula summaries, a copula KL divergence (CKL) and a copula entropy gap (CED), and establish basic consistency and central limit results for their plug-in estimators. In controlled experiments, CD reliably separates on-target and off-target copulas with matched Kendall's , provides a dependence-aware signal for tuning SGLD step sizes where Effective Sample Size favors overly aggressive (and biased) settings, and remains stably nonzero under deliberate tail-dependence mismatch where a naive -based diagnostic fails; CKL and CED offer a complementary Shannon-style view that echoes these findings. Timing benchmarks show that both CD variants incur only millisecond-level overhead over the tested range and exhibit near-linear empirical scaling in sample size, providing a lightweight, dependence-focused complement to quadratic-cost omnibus discrepancies such as the Kernel Stein Discrepancy (KSD).

Paper Structure

This paper contains 55 sections, 12 theorems, 110 equations, 7 figures, 7 tables, 3 algorithms.

Table of Contents

  1. INTRODUCTION
  2. RELATED WORK
  3. MCMC diagnostics and Stein's method
  4. Biased MCMC methods
  5. Copula theory and applications
  6. Dependence, tail behavior, and copula comparison
  7. When to use CD versus Stein/KSD.
  8. NOTATION
  9. Distributions and samples.
  10. Copula framework.
  11. Pseudo-observations and ties.
  12. Estimators and discrepancies.
  13. Shannon-style copula functionals.
  14. Other diagnostics.
  15. THE COPULA DISCREPANCY FRAMEWORK
  16. ...and 40 more sections

Key Result

Theorem 1

Let $\{(U_i,V_i)\}_{i=1}^n$ be i.i.d. pseudo-observations from a copula $C_{\theta_P}$ in the target family. Let $\hat{\tau}_n$ be the sample Kendall's tau and $\hat{\theta}_n^{(M)}=\tau^{-1}(\hat{\tau}_n)$ be the moment-based estimator. Under Assumption assump:regularity:

Figures (7)

  • Figure 1: MLE-based Copula Discrepancy (CD) for samples drawn from the on-target Gumbel copula (blue) and the off-target Clayton copula (orange), both constructed to have the same population Kendall’s tau $\tau = 0.6$. Each point is the mean CD over 100 replications; the shaded bands show the 95% confidence interval for the mean. Even though rank correlation is matched, the on-target CD decays rapidly toward zero as $n$ increases, while the off-target CD remains substantially larger (around $8\times 10^{-2}$ to $10^{-1}$) and varies only mildly across sample sizes.
  • Figure 2: Information–theoretic diagnostics for Experiment \ref{['sec:exp1']}. Left: Copula KL-based discrepancy (CKL) evaluated against a Kendall’s-tau-matched Gumbel reference copula. Right: Copula entropy gap (CED) between the fitted copula and the Gumbel target. In both panels, blue corresponds to on-target Gumbel samples and orange to off-target Clayton samples; lines show means over 100 replications and shaded bands show 95% confidence intervals. CKL and CED reproduce the same qualitative ordering as CD: fits to on-target samples trend toward lower discrepancy as $n$ grows, while fits to off-target samples remain consistently higher.
  • Figure 3: Hyperparameter selection for SGLD targeting a bimodal Gaussian mixture. Left: Mean Copula Discrepancy (CD, solid line; left y-axis, log$_{10}$ scale) and Mean Effective Sample Size (ESS, dashed line; right y-axis) as functions of the SGLD step-size $\epsilon$. Shaded bands show 95% confidence intervals for the means over 100 replications (computed on the original scale and displayed after transformation for CD). ESS is ultimately maximized at the largest step-size ($\epsilon = 10^{-1}$), but is not strictly monotone over the full grid, with a small dip in the smallest-$\epsilon$ regime. In contrast, the CD curve attains its minimum at an intermediate step-size ($\epsilon = 4.64\times 10^{-3}$), indicating improved agreement with the target dependence structure at this setting. Right: SGLD samples for the ESS-selected step-size ($\epsilon = 10^{-1}$, blue triangles) and the CD-selected step-size ($\epsilon = 4.64\times 10^{-3}$, orange circles). The ESS-selected sample cloud is visibly more dispersed around each mode, while the CD-selected configuration yields a tighter concentration around the two modal regions.
  • Figure 4: Shannon-type dependence diagnostics for the SGLD hyperparameter study. Left: Copula KL (CKL, forward KL) estimate versus step-size $\epsilon$, with mean and 95% confidence interval over 100 replications. Right: Copula Entropy Gap (CED) versus $\epsilon$, again with mean and 95% confidence interval. Both summaries vary with $\epsilon$ and agree with CD in preferring an intermediate regime; in particular, both CKL and CED attain their lowest means at $\epsilon = 4.64\times 10^{-3}$. However, their uncertainty bands remain wide relative to the scale of changes across $\epsilon$, so the CD plot in Figure \ref{['fig:exp2']} remains the most directly interpretable diagnostic for selecting a step-size in this setting.
  • Figure 5: Experiment 3: diagnostics for a tail-dependence mismatch. The Naive Tau Discrepancy (orange, left axis) shrinks toward zero as $n$ increases and would incorrectly suggest that the Gumbel sample is compatible with the Clayton target, since their Kendall's taus agree by construction. In contrast, the MLE-based Copula Discrepancy (CD, blue circles, left axis) remains bounded away from zero across all sample sizes, indicating a persistent structural mismatch driven by tail dependence. The KSD (blue triangles, right axis) also detects the mismatch and increases with sample size, but it lives on a different scale and is therefore not directly comparable in magnitude to CD. Both axes are shown on logarithmic scales. Shaded regions show 95% confidence intervals for the mean over 100 replications.
  • ...and 2 more figures

Theorems & Definitions (22)

  • Remark 3.1: Moment-based CD as a baseline
  • Definition 4.1: Archimedean copulas
  • Definition 4.2: Copula Discrepancy
  • Remark 4.1: Intuition under misspecification
  • Theorem 1: Consistency of Moment-Based Estimator
  • Theorem 2: Asymptotic Distribution of Moment-Based CD
  • Corollary 4.1: Asymptotic level-$\alpha$ test based on moment-based CD
  • Definition 4.3: Copula equivalence test
  • Theorem 3: Asymptotic test and rejection rule
  • Theorem 4: Bounded influence for the moment-based CD
  • ...and 12 more