Conditional Negative Association

• Conditional Negative Association is a property where negative dependence between disjoint coordinate blocks persists even after conditioning on any subset.
• In competing-urn models, CNA is established using techniques like ratio monotonicity, strong log-concavity, and graph orientation to demonstrate stochastic antagonism.
• This framework distinguishes itself from standard negative association by ensuring that conditional measures remain negatively associated, even under complex thresholding.

Conditional negative association (CNA) is a strengthening of negative association for measures on finite product spaces: after conditioning on any specified subset of coordinates, the remaining coordinates still satisfy the negative dependence inequality for increasing functions on disjoint coordinate blocks. In the material considered here, the clearest classical realization is Kahn and Neiman’s analysis of competing urns, where threshold urn measures are proved to be CNA and, in the i.i.d. placement case, the result extends to interval-valued occupancy variables (Kahn et al., 2010).

1. Formal definition and basic variants

Let $\mu$ be a probability measure on a finite product space

$\Omega=\prod_{i=1}^n \Omega_i,$

where each $\Omega_i$ is a chain. Two events $A,B$ are negatively correlated, written $A\downarrow B$, if

$\Pr(A\cap B)\le \Pr(A)\Pr(B).$

For real-valued random variables $X,Y$, the notation $X\downarrow Y$ means

$$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$

equivalently,

$$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$

for all increasing $\Omega=\prod_{i=1}^n \Omega_i,$0 (Kahn et al., 2010).

Negative association (NA) requires this inequality for increasing observables on disjoint coordinate sets. If $\Omega=\prod_{i=1}^n \Omega_i,$1 and $\Omega=\prod_{i=1}^n \Omega_i,$2 are increasing events depending on disjoint coordinate sets, then NA means $\Omega=\prod_{i=1}^n \Omega_i,$3. In the $\Omega=\prod_{i=1}^n \Omega_i,$4 setting, the weaker pairwise property is negative correlation (NC), namely $\Omega=\prod_{i=1}^n \Omega_i,$5 for all $\Omega=\prod_{i=1}^n \Omega_i,$6. Conditional negative association strengthens NA by requiring that every law obtained by conditioning on some coordinates remains NA. If $\Omega=\prod_{i=1}^n \Omega_i,$7, then CNA means that after conditioning on any event of the form

$\Omega=\prod_{i=1}^n \Omega_i,$8

with positive probability, the remaining coordinates satisfy

$\Omega=\prod_{i=1}^n \Omega_i,$9

for all increasing $\Omega_i$0 depending on disjoint subsets of coordinates outside $\Omega_i$1 (Kahn et al., 2010).

Two nearby notions are standard. Conditional negative correlation (CNC) is the pairwise analogue of CNA. In the binary strong-Rayleigh literature, one also encounters a stronger closure notion denoted $\Omega_i$2: a measure is $\Omega_i$3 if every measure obtained by conditioning on some coordinates is NA, and $\Omega_i$4 if every measure obtained by imposing external fields and projections is $\Omega_i$5 (Günter et al., 2018).

2. Competing urns as the canonical classical model

The competing-urns model starts from a random map

$\Omega_i$6

where the variables $\Omega_i$7 are independent. In the ordinary model they are i.i.d.; in the generalized model they may have different laws. The occupancy count of urn $\Omega_i$8 is

$\Omega_i$9

Given thresholds $A,B$0, the threshold indicators are

$A,B$1

More generally, for each $A,B$2, choose

$A,B$3

and define interval variables by

$A,B$4

The law of $A,B$5 is then a threshold urn measure in the threshold case, or an interval urn measure in the more general case (Kahn et al., 2010).

The principal urn theorem is:

Theorem 1. Threshold urn measures are CNA.

A stronger structural result yields it:

Theorem 2. If the $A,B$6’s are i.i.d. then the $A,B$7’s in the interval-valued construction above are CNA.

Accordingly, the final CNA conclusion is not restricted to exceedance indicators $A,B$8; in the i.i.d. case it holds for interval-valued occupancy discretizations as well. At the same time, the generalized independent-but-nonidentical model is handled only partially: several monotonicity and log-concavity statements are established there, implying CNC and NA in various forms, but not full CNA. Whether generalized urn, threshold urn, or interval urn measures are CNA is left open as Question 13 (Kahn et al., 2010).

This model isolates the central difficulty of CNA. The hard identity $A,B$9 makes global competition obvious, but CNA asks whether that antagonism survives coarse thresholding and arbitrary coordinate conditioning. The urn theorems show that, in the i.i.d. setting, it does.

3. Reduction to two-block antitonicity

The proof of CNA in competing urns is organized around a two-block conditional negative correlation statement. Partition the urn set as $A\downarrow B$0, fix interval values $A\downarrow B$1 for $A\downarrow B$2, and condition on

$A\downarrow B$3

Then define

$A\downarrow B$4

The key claim is

$A\downarrow B$5

From this, NA under the conditioning $A\downarrow B$6 follows, and hence CNA (Kahn et al., 2010).

The reduction works as follows. If $A\downarrow B$7 and $A\downarrow B$8 are increasing events determined by the $A\downarrow B$9-coordinates in $\Pr(A\cap B)\le \Pr(A)\Pr(B).$0 and $\Pr(A\cap B)\le \Pr(A)\Pr(B).$1, define

$\Pr(A\cap B)\le \Pr(A)\Pr(B).$2

A coupling shows $\Pr(A\cap B)\le \Pr(A)\Pr(B).$3 and $\Pr(A\cap B)\le \Pr(A)\Pr(B).$4 are increasing, and conditional on $\Pr(A\cap B)\le \Pr(A)\Pr(B).$5, the events $\Pr(A\cap B)\le \Pr(A)\Pr(B).$6 and $\Pr(A\cap B)\le \Pr(A)\Pr(B).$7 are independent under $\Pr(A\cap B)\le \Pr(A)\Pr(B).$8. Therefore

$\Pr(A\cap B)\le \Pr(A)\Pr(B).$9

which is exactly the NA inequality after conditioning on the coordinates in $X,Y$0 (Kahn et al., 2010).

The central probabilistic input is stronger than mere negative correlation. Writing

$X,Y$1

the paper proves the ratio-monotonicity inequality

$X,Y$2

whenever the ratios are defined, together with convexity of the support

$X,Y$3

These two facts imply that $X,Y$4 is stochastically decreasing in $X,Y$5, hence $X,Y$6 given $X,Y$7 (Kahn et al., 2010).

A second aggregate variable,

$X,Y$8

becomes decisive in the i.i.d. case. Conditional on $X,Y$9,

$X\downarrow Y$0

where $X\downarrow Y$1 is the total probability of landing in $X\downarrow Y$2 relative to $X\downarrow Y$3. Thus sufficient log-concavity of the law of $X\downarrow Y$4 forces antagonism between $X\downarrow Y$5 and $X\downarrow Y$6. The paper proves that the law of $X\downarrow Y$7 is strongly log-concave and strengthens this further to ultra-log-concavity, which in the i.i.d. case closes the argument from two-block antitonicity to full CNA (Kahn et al., 2010).

4. Structural inequalities, support convexity, and graph orientations

The inductive engine behind the urn results is Theorem 4. Singling out $X\downarrow Y$8, define

$X\downarrow Y$9

Then the theorem states:

• $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$0 is nonincreasing in $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$1;
• $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$2

Part (b) says precisely that $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$3 is strongly log-concave. Part (a) is the occupancy-ratio monotonicity from which the two-block comparison is derived. The proof uses induction on the number of balls, together with the decomposition

$$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$4

for any event $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$5 determined by the occupancies of urns other than $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$6, and the elementary ratio bound

$$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$7

A support-convexity proposition then removes the possibility of internal gaps (Kahn et al., 2010).

That support-convexity proposition is itself graph-theoretic. If two realizations $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$8 have positive probability and one urn receives more balls under $$\{X\ge s\}\downarrow \{Y\ge t\}\qquad \forall s,t\in\mathbb R,$$9 than under $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$0, an alternating-path rerouting in the bipartite incidence graph of balls and urns moves one ball while staying inside the support. The consequence is convexity of the support of occupancy vectors under interval constraints (Kahn et al., 2010).

A separate combinatorial pillar is Lemma 8 on graph orientations. For a multigraph $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$1 on vertex set $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$2, let $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$3 be the set of orientations satisfying

$$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$4

and let $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$5. If $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$6 satisfy

$$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$7

then

$$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$8

The lemma is used in three ways: to give another proof of the ratio inequalities; to upgrade strong log-concavity to ultra-log-concavity for the law of $$\mathbb E[f(X)g(Y)]\le \mathbb E[f(X)]\,\mathbb E[g(Y)]$$9; and to yield a negative lattice condition

$\Omega=\prod_{i=1}^n \Omega_i,$00

for certain occupancy-derived set functions. The same orientation argument also gives

$\Omega=\prod_{i=1}^n \Omega_i,$01

when $\Omega=\prod_{i=1}^n \Omega_i,$02, and Theorem 12 proves ultra-log-concavity for the number of valid partial $\Omega=\prod_{i=1}^n \Omega_i,$03-maps under lower and upper degree constraints (Kahn et al., 2010).

These results show that CNA in competing urns is not derived from a single covariance estimate. It rests on a combination of interval-conditioned occupancy ratios, convex support geometry, and a graph-orientation monotonicity principle.

5. Relation to adjacent negative-dependence frameworks

The urn results sit within a broader hierarchy of negative dependence notions. In the same competing-urns setting, Dubhashi and Ranjan had already shown that the finer placement indicators

$\Omega=\prod_{i=1}^n \Omega_i,$04

are NA even for nonidentical balls. That implies NA for generalized threshold urn measures, since threshold events are increasing functions of disjoint blocks of the $\Omega=\prod_{i=1}^n \Omega_i,$05’s. What it does not give is CNA after fixing some threshold coordinates, because conditioning destroys the block-product structure. The contribution of Kahn and Neiman is precisely to control that conditional behavior (Kahn et al., 2010).

Other negative-dependence frameworks give comparison points rather than direct CNA theorems. Pairwise counter-monotonicity is an extremal dependence structure under which every pair of coordinates is counter-monotonic; it implies NA because increasing functions of disjoint coordinate blocks remain pairwise counter-monotonic, after which the Fréchet–Hoeffding inequality yields the covariance bound. That work does not define or study CNA, but it clarifies one route by which strong sample-path antagonism can imply ordinary NA (Lauzier et al., 2023).

In sequential Monte Carlo, a closely related but different pattern appears. The resampling analysis of offspring counts is carried out conditionally on the current particle system, and the main consistency theorem assumes that for every fixed input $\Omega=\prod_{i=1}^n \Omega_i,$06, the count vector $\Omega=\prod_{i=1}^n \Omega_i,$07 is NA. This is a fiberwise, conditional-on-the-input NA hypothesis rather than an abstract closure theorem saying that NA is preserved under conditioning (Gerber et al., 2017).

Algorithmic dependent rounding exhibits yet another variant. For the dependent-rounding output $\Omega=\prod_{i=1}^n \Omega_i,$08, the appendix of the harmonic $\Omega=\prod_{i=1}^n \Omega_i,$09-median / minimization-PAV paper defines CNA explicitly by requiring that

$\Omega=\prod_{i=1}^n \Omega_i,$10

be NA for every subset $\Omega=\prod_{i=1}^n \Omega_i,$11 and assignment $\Omega=\prod_{i=1}^n \Omega_i,$12. It further explains that carefully implemented dependent rounding on a fixed tournament tree yields NA/CNA, while the paper’s own analysis uses a weaker binary negative association property to obtain concentration for conditional random variables (Byrka et al., 2017).

For random measures and point processes, finite-dimensional NA of disjoint count vectors implies NA of the full random measure, and NA is preserved under weak convergence. That framework is technically important for extending NA beyond finite coordinate systems, but it does not itself develop a point-process theory of CNA; the only explicit CNA terminology there comes from cited binary-measure results in the strong-Rayleigh setting (Last et al., 2019).

6. Scope, limitations, conjectures, and terminological ambiguities

The principal scope limitation in the competing-urns theory is the nonidentical-ball case. Full CNA is proved only when the placements are i.i.d. The ratio inequalities, support convexity, and strong log-concavity statements survive for independent nonidentically distributed balls, and they imply several conditional negative dependence properties, including CNC and interval-conditioned monotonicity, but the final step to CNA relies on the i.i.d. binomial splitting of $\Omega=\prod_{i=1}^n \Omega_i,$13 given $\Omega=\prod_{i=1}^n \Omega_i,$14 (Kahn et al., 2010).

This gap is tied to several open conjectural relations. The paper asks whether generalized urn, generalized threshold urn, or generalized interval urn measures are CNA. It also notes that a negative answer would disprove a conjecture of Pemantle that CNC implies CNA. Conversely, Theorem 1 provides a natural counterexample to the conjecture “CNA implies ULC,” because ordinary urn measures are not always ULC even though threshold urn measures are CNA (Kahn et al., 2010).

The same paper also situates CNA among older conjectures of Farr and Welsh. For a decreasing family $\Omega=\prod_{i=1}^n \Omega_i,$15, with

$\Omega=\prod_{i=1}^n \Omega_i,$16

Farr conjectured conditional negative correlation for a graph-independent-set family $\Omega=\prod_{i=1}^n \Omega_i,$17, and Welsh conjectured the same for arbitrary decreasing $\Omega=\prod_{i=1}^n \Omega_i,$18. Kahn and Neiman give a counterexample to Welsh’s stronger claim. This does not conflict with the urn CNA theorem because the events $\Omega=\prod_{i=1}^n \Omega_i,$19 in that counterexample are far more general than threshold occupancy events (Kahn et al., 2010).

A separate source of confusion is terminological. In association-rule mining, “conditional negative association” can refer informally to rules such as

$\Omega=\prod_{i=1}^n \Omega_i,$20

with large

$\Omega=\prod_{i=1}^n \Omega_i,$21

together with support and interest thresholds. That usage concerns infrequent itemsets and dependence from transactional data; it is not the probabilistic CNA property on product measures (Kong et al., 2018). Likewise, work on quantum channels that destroy negative conditional entropy studies the quantum-information-theoretic condition

$\Omega=\prod_{i=1}^n \Omega_i,$22

which is unrelated to classical negative association despite the similar phrase “negative conditional” (Srinidhi et al., 2023).

In its classical probabilistic meaning, CNA is therefore best understood as a conditioning-stable form of NA. The competing-urns results show that this stability can be proved in a nontrivial occupancy model by combining interval-conditioned stochastic antitonicity, strong log-concavity, and graph-theoretic monotonicity, while the surrounding literature makes clear that neither unconditional NA nor other adjacent negative-dependence notions automatically provide such a conditioning-stable theory (Kahn et al., 2010).