Global Dependence Order Explained

• Global dependence order is a framework for ranking input variables by the probabilistic strength of their dependence with a response, defined through clear axioms and invariance properties.
• The conditional convex order quantifies dependence using conditional survival probabilities and convex functions, ensuring robust invariance under transformations and data processing.
• Empirical global dependence orders use sensitivity indices derived from divergence measures to outperform classical variance-based methods in nonlinear and high-dimensional models.

Global dependence order refers to formal mechanisms for ranking or ordering random variables or multivariate predictors according to the “strength” of their probabilistic dependence with a response variable. Approaches to global dependence ordering form crucial underpinnings in sensitivity analysis, model interpretability, and functional data analysis, enabling both principled comparison of input-output relationships and the calibration of dependence measures against explicit axioms. Two major frameworks embodying the concept are the conditional convex order (as a strict global order on functional dependence, satisfying a full suite of axioms) (Ansari et al., 9 Nov 2025), and empirical global dependence orders based on sensitivity indices derived from dissimilarity-based dependence measures (Veiga, 2013). Both frameworks provide invariance properties, algorithmic ranking procedures, and operational interpretations in statistical modeling and machine learning.

1. The Conditional Convex Order as a Global Dependence Order

Let $(Y, X)$ and $(Y', X')$ be random vectors, with $Y, Y'$ real-valued and $X, X'$ having (potentially different) dimensions. The global dependence order, called the conditional convex order (“ccx-order”), is defined as follows:

• For $v \in (0,1)$, let $q_Y(v) = \inf \{y: F_Y(y) \geq v\}$ be the generalized inverse of the CDF $F_Y$, and denote $\eta_{Y|X}^v := \mathbb{P}(Y \geq q_Y(v)\;|\;X)$.
• Then $(Y, X) \preceq_{ccx} (Y', X')$ if for every $v \in (0,1)$,

$\eta_{Y|X}^v \leq_{cx} \eta_{Y'|X'}^v,$

meaning that for every convex function $\varphi : [0,1] \to \mathbb{R}$,

$$\mathbb{E}[\varphi(\eta_{Y|X}^v)] \leq \mathbb{E}[\varphi(\eta_{Y'|X'}^v)].$$

This order compares the “spread” of conditional survival probabilities across predictors, capturing their joint ability to localize $Y$ given $X$. The approach is copula-invariant and compatible with dimension-reduction schemes.

2. Axiomatic Characterization: The Eight Axioms

A relation $\preceq$ on pairs $(Y, X)$ is a global dependence order if and only if it satisfies the following axioms:

Code	Name	Description
O1	Law-invariance	If $(Y, X)$, $(Y', X')$ have identical joint laws, then they are equivalent in order: $(Y, X) \approx (Y', X')$.
O2	Quasi-ordering	Reflexivity and transitivity hold: $\preceq$ is a quasi-order.
O3	Independence as unique minimum	$(Y, X) \preceq (Y', X')$ for all $(Y', X')$ if and only if $X \perp Y$.
O4	Perfect-dependence as unique maximum	$(Y', X') \preceq (Y, X)$ for all $(Y', X')$ if and only if $Y = f(X)$ a.s. for some measurable $f$.
O5	Information monotonicity (data proc.)	For any predictor enlargement $Z$, $(Y, X) \preceq (Y, (X, Z))$.
O6	Conditional independence char.	$(Y, X) \approx (Y, (X, Z))$ if and only if $Y \perp Z \mid X$.
O7	Transformation invariance	$(Y, X) \approx (g(Y), h(X))$ for any strictly increasing $g$, bijection $h$.
O8	Copula (marginal) invariance	Replacing $Y$ by $F_Y(Y)$ or $X$ by its marginal PIT leaves the order unchanged.

These axioms uniquely determine the ccx-order among all possible orders of dependence strength. Of note is the information monotonicity (O5), which ensures that adding redundant predictors cannot decrease perceived dependence, and the exclusive conditional-independence characterization (O6), unattained in other orderings.

3. Extremal Elements: Independence and Perfect Dependence

The conditional convex order realizes independence and deterministic functional dependence as the universal minimum and maximum, respectively:

• Minimal (Independence): For all $v$, $\eta_{Y|X}^v$ is constant (unaffected by $X$), so $(Y, X)$ is minimal.
• Maximal (Functional dependence): For all $v$, $\eta_{Y|X}^v$ takes values only in $\{0,1\}$ a.s.; such two-point conditional variables are maximal in convex order.

This guarantees that $(Y, X)$ with $Y=f(X)$ is strictly greater than any partially dependent or independent system.

4. Fundamental Properties: Conditional Independence, Invariance, Data-processing

Conditional Independence and Data-processing

For any $(Y, X, Z)$, the order guarantees $(Y, X) \approx (Y, (X, Z))$ if and only if $Y \perp Z\,|\,X$, making ccx-order uniquely sensitive to all conditional independences.

Data-Processing Property: For any additional predictor $Z$, $(Y, X) \preceq (Y, (X, Z))$. If $Z$ adds no information about $Y$ given $X$, equality holds.

Invariance

• Monotone Recoding: Strictly increasing $g$ on $Y$’s support and bijections on $X$ do not affect order.
• Copula Invariance: Order is invariant under marginal probability integral transforms, highlighting the intrinsic copula-level comparison of the dependence structure.

5. Relationship to Classical Orders

Schur Order and Concordance Order

• Schur Order: The ccx-order is equivalent to a Schur ordering of conditional CDFs:

$u \mapsto \eta_{Y|X}^v(q_X(u))$

is Schur-smaller than that for $(Y', X')$, i.e., has more concentrated partial integrals, in the sense of Hardy–Littlewood–Pólya.

• Concordance Order ($\leq_c$): For bivariate stochastically increasing (SI) models with continuous margins, $(Y, X) \preceq_{ccx} (Y', X')$ coincides with $(Y, X) \leq_c (Y', X')$, i.e., the pointwise order on joint CDFs and survival functions. More generally, concordance order’s extrema are comonotonic/anti-comonotonic copulas, whereas ccx-order’s maxima and minima are strict independence and functional dependence.

6. Instantiation in Key Model Classes

Additive Error Models

For $Y=f(X)+\sigma \epsilon$, $\epsilon \perp X$, the ccx-order quantifies the effect of noise:

• If $0 \leq \sigma < \sigma'$, then $(Y, X) \succeq_{ccx} (Y', X)$.
• Any dependence measure increasing in ccx-order (e.g., Chatterjee’s $\xi$) is decreasing in $\sigma$.

Multivariate Normal Distributions

For $(Y, X) \sim \mathcal{N}(\mu, \Sigma)$,

$$(Y, X) \preceq_{ccx} (Y', X') \iff \Sigma_{Y,X} \Sigma_X^{-1} \Sigma_{X,Y}/\sigma_Y^2 \leq \Sigma'_{Y,X} {\Sigma'_X}^{-1} \Sigma'_{X,Y}/\sigma_{Y'}^2$$

The classical fraction of explained variance $$\operatorname{Var}[\mathbb{E}(Y|X)]/\operatorname{Var}(Y)$$ is monotone in the ccx-order.

Copula-Based Models

For $Y \perp X\, |\, g(X)$ and $Y' \perp X' \mid h(X')$, if the bivariate copulas $C_{Y, g(X)}$ and $C_{Y', h(X')}$ lie within a stochastically increasing family ordered by concordance, then $C_{Y, g(X)} \leq_c C_{Y', h(X')}$ implies $(Y, X) \preceq_{ccx} (Y', X')$.

7. Dependence Measures and the Global Dependence Order

Chatterjee’s $\xi$

The rank correlation $\xi(Y, X)$ is defined as

$$\xi(Y, X) = \frac{\int \operatorname{Var}(\mathbb{P}(Y \geq y\,|\,X))\, dP^Y(y)}{\int \operatorname{Var}(1_{Y \geq y})\, dP^Y(y)}$$

Alternatively, $$\xi(Y, X) \propto \int [\mathbb{E}(\eta_{Y|X}^v)^2]\, dv - \mathrm{const}$$. It is monotonic with respect to the ccx-order.

Generalized $\phi$-functionals

For any convex $\phi$, define

$$\xi_{\phi}(Y, X) = \alpha_{\phi}^{-1} \int \int \phi(F_{Y|X=x}(y) - F_Y(y))\, dP^X(x) dP^Y(y)$$

Two-point “sensitivity” versions $\Lambda_{\phi}$ similarly compare $\phi$ of pairwise differences of conditional CDFs. All such measures are ccx-increasing, vanishing only at independence, and attaining their maximum at perfect dependence.

Rearranged Concordance-Based Measures

For any copula-based concordance measure $\mu$ on SI vectors (such as Spearman’s $\rho$, Gini’s $\gamma$), define

$$\mu(Y, X) := \mu(F_Y \circ q_{Y|X}^{\uparrow}(V), U)$$

where $q_{Y|X}^{\uparrow}$ is the increasing rearrangement by $V \sim \mathrm{Uniform}(0,1)$. All such $\mu$ measures are ccx-increasing, characterize independence and perfect dependence, and possess information-monotonicity.

8. Empirical Global Dependence Orders from Sensitivity Indices

Alternative approaches establish a global dependence order on input variables via sensitivity index-based ranking (Veiga, 2013):

Let $Y = \eta(X^1, \ldots, X^p)$, with independent $X^k$ and known law. For each input $X^k$, define:

$S_k := \mathbb{E}_{X^k} [ d(Y,\,Y|X^k) ]$

where $d(\cdot, \cdot)$ is a suitable dissimilarity or divergence. Key realizations include:

• f-divergence indices: Use convex $f$ to measure divergences between $p_Y$ and $p_{Y|X^k=x}$.
  • Borgonovo’s index: $f(t) = |t-1|$
  • Mutual information: $f(t) = t\ln t$
  • Pearson $\chi^2$: $f(t) = (t-1)^2/t$
• Distance correlation ($\mathrm{dCor}$): Characteristic function–based quadratic integral, normalized to $[0,1]$.
• Hilbert–Schmidt Independence Criterion (HSIC): Kernel-trace based covariance on RKHS.

A total ordering results from ranking $S_k$ (or its estimator) across variables. Procedures include sorting, mRMR for redundancy-penalized selection, or feature selection via HSIC-Lasso.

9. Computation and Practical Illustration

For f-divergence indices, density ratio estimation (via unconstrained least-squares importance fitting or ML) is used. dCor and HSIC are both $O(n^2)$ in sample size and provide pseudocode for efficient computation. All indices are:

• Transformation invariant (to varying degrees)
• Non-additive (do not sum to one as variance-based Sobol indices)
• Consistent as $n \to \infty$
• Able to handle multivariate $X^k$ and $Y$

Empirical global dependence orders outperform Sobol indices in nonlinear, interaction-heavy, or high-dimensional settings, as in the Ishigami and Morris models or industrial cases (e.g., Punq reservoir, Marthe maps). This suggests their robustness in screening influential inputs under minimal structural assumptions.

10. Summary and Outlook

A global dependence order provides a law-invariant, data-processing-respecting framework to order (possibly multivariate) variables by their dependence strength with an output. The conditional convex order is characterized by a unique set of axioms and connects to Schur and concordance orderings, while also providing operational semantics for a variety of dependence measures. Empirical approaches based on generalized sensitivity indices yield practical ordering and screening tools in complex, high-dimensional, and nonlinear models, underscoring the central role of robust dependence quantification in both theory and application.