Bounded Statistical Parity: Concepts & Trade-offs

• Bounded statistical parity is a relaxation of demographic parity that limits the dependence between outputs and sensitive attributes via global (mutual information) or local (χ² divergence) constraints.
• It unifies fairness, privacy, and compression by balancing utility trade-offs through controlled leakage bounds in representation learning and classification settings.
• Key methodologies include using extended Functional Representation Lemmas and quadratic approximations to analyze trade-offs with base-rate imbalances and equalized odds.

Bounded statistical parity denotes a relaxation of demographic or statistical parity in which exact independence between an output and a sensitive attribute is replaced by an explicit upper bound on dependence or disparity. In the information-theoretic literature, two formalizations are central: a global leakage budget $I(Y;S)\le \epsilon$, which bounds the mutual information between a representation $Y$ and a sensitive attribute $S$, and a local, point-wise constraint $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$ for every realized output $y$, which bounds posterior deviation from the population prior at each output (Zamani et al., 18 Aug 2025, Zamani et al., 27 Nov 2025). In classification settings, the same theme appears as a bounded acceptance-rate disparity, typically of the form $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$, with the achievable bound determined by base-rate imbalance and error-rate structure (Bargh et al., 26 Jan 2026).

1. Definitions and formal variants

Classical statistical parity, also called demographic parity, requires that the algorithm’s output be independent of the sensitive attribute. In a representation-learning formulation this is written as $Y\perp S$, equivalently $I(Y;S)=0$, and in a binary prediction setting it is written as

$Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$

The first form emphasizes distributional independence; the second emphasizes equality of positive prediction rates across groups (Zamani et al., 18 Aug 2025, Bargh et al., 26 Jan 2026).

Bounded statistical parity replaces this exact condition by a controlled relaxation. In the mutual-information formulation,

$I(Y;S)\le \epsilon,$

where $Y$0 is a leakage or fairness budget. The case $Y$1 recovers perfect statistical parity or perfect privacy, while $Y$2 permits small, controlled dependence between $Y$3 and $Y$4 (Zamani et al., 18 Aug 2025). In the point-wise formulation, fairness is enforced for each output realization: $Y$5 This requires the posterior distribution of the sensitive attribute conditioned on a specific output $Y$6 to remain close to the marginal $Y$7 (Zamani et al., 27 Nov 2025).

The two formulations differ in granularity. Mutual information is a global measure of leakage, whereas the point-wise $Y$8 bound controls local posterior deviations for every output. The latter is therefore stronger than an average leakage constraint; the point-wise paper explicitly states that its achievable utility $Y$9 is bounded above by the utility achievable under the mutual-information relaxation $S$0 (Zamani et al., 27 Nov 2025).

Formulation	Constraint	Interpretation
Exact statistical parity	$S$1 or $S$2	Perfect independence
Bounded statistical parity	$S$3	Global leakage budget
Bounded point-wise statistical parity	$S$4	Per-output posterior control

A broader interpretation, stated explicitly in the point-wise work, is that bounded statistical parity refers to imposing an explicit upper bound on deviations from demographic parity rather than requiring exact independence (Zamani et al., 27 Nov 2025).

2. Representation-theoretic problem formulations

The information-theoretic literature studies bounded statistical parity through representation design. The canonical variables are $S$5 for the sensitive attribute or secret, $S$6 for useful data, $S$7 for the task variable, and $S$8 for the designed representation (Zamani et al., 18 Aug 2025, Zamani et al., 27 Nov 2025).

One formulation allows the encoder to depend on all variables through

$S$9

with no requirement that $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$0 hold. The optimization problem is

$\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$1

Its three terms have fixed roles: $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$2 is utility, $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$3 is a compression or encoding-rate constraint, and $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$4 is the bounded statistical parity or bounded privacy leakage constraint (Zamani et al., 18 Aug 2025).

A second formulation assumes the agent has no direct access to $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$5 or $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$6, only to $\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$7. The encoder is then

$\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$8

and the Markov chains

$\chi^2(P_{S\mid Y=y};P_S)\le \epsilon^2$9

are imposed. The corresponding optimization problem is

$y$0

Here again, the objective is utility, the rate $y$1 controls compression, and $y$2 controls allowed deviation from demographic parity (Zamani et al., 27 Nov 2025).

Both formulations explicitly unify fairness, privacy, and compression. The first paper also states that the problem can be interpreted as one of code design with bounded leakage and bounded rate, treating $y$3 as a secret and $y$4 as a codeword whose leakage about $y$5 and rate with respect to $y$6 are simultaneously controlled (Zamani et al., 18 Aug 2025).

3. Mutual-information bounded parity

The mutual-information formulation treats bounded statistical parity as a global information budget: $y$7 This is also a bounded privacy leakage condition, since $y$8 measures how much information the representation reveals about the sensitive attribute (Zamani et al., 18 Aug 2025).

A central result is the general upper bound

$y$9

The two terms correspond to relaxing one constraint at a time: if only the fairness constraint remains, utility is bounded by $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$0; if only the rate constraint remains, utility is bounded by $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$1 (Zamani et al., 18 Aug 2025).

The same paper develops lower bounds using extended versions of the Functional Representation Lemma and the Strong Functional Representation Lemma. The relevant tools are the Extended FRL, which constructs an auxiliary variable $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$2 satisfying $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$3 and $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$4, and the Extended SFRL, which adds a bound on $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$5 with

$$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$6

These extensions support constructive mechanisms based on randomization over $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$7 or $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$8 and yield regime-dependent lower bounds such as $$|Pr(\hat Y=1\mid S=1)-Pr(\hat Y=1\mid S=0)|\le \varepsilon$$9, $Y\perp S$0, $Y\perp S$1, and $Y\perp S$2 (Zamani et al., 18 Aug 2025).

The framework also clarifies why strict parity can be too restrictive. Under the structural conditions that $Y\perp S$3 holds and $Y\perp S$4 is invertible, the paper’s Example 1 shows that perfect parity $Y\perp S$5 implies $Y\perp S$6, and then $Y\perp S$7. In that regime, exact demographic parity forces zero utility. The paper uses this example to motivate bounded rather than perfect parity (Zamani et al., 18 Aug 2025).

Two numerical examples illustrate the resulting trade-offs. In a quantized-histogram setting derived from grayscale digit images, with a ternary sensitive attribute $Y\perp S$8, the lower bound $Y\perp S$9 matches the upper bound $I(Y;S)=0$0 for $I(Y;S)=0$1, yielding a tight characterization for sufficiently large $I(Y;S)=0$2. In a noisy typewriter example with $I(Y;S)=0$3 uniform on $I(Y;S)=0$4 and a deterministic ternary $I(Y;S)=0$5, the SFRL-based lower bound $I(Y;S)=0$6 is tighter than $I(Y;S)=0$7 in the regime $I(Y;S)=0$8, while $I(Y;S)=0$9 becomes the active upper bound for $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$0 (Zamani et al., 18 Aug 2025).

The main conceptual conclusion is explicit in the paper: allowing non-zero leakage enlarges the feasible region and can improve the attained utility. The trade-off curves between $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$1 and $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$2 shift upward as $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$3 increases (Zamani et al., 18 Aug 2025).

4. Point-wise bounded parity and information geometry

The point-wise formulation imposes fairness locally: $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$4 The divergence is defined as a scaled squared Euclidean distance between the posterior vector $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$5 and the prior $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$6, with scaling by $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$7. If $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$8, the divergence is zero; larger values indicate greater deviation from statistical parity at output $Pr(\hat Y=1\mid S=1)=Pr(\hat Y=1\mid S=0).$9 (Zamani et al., 27 Nov 2025).

The paper studies the high-fairness regime in which $I(Y;S)\le \epsilon,$0 is small. It parameterizes the posterior as

$I(Y;S)\le \epsilon,$1

where $I(Y;S)\le \epsilon,$2 satisfies normalization, marginal consistency, and a unit $I(Y;S)\le \epsilon,$3-norm constraint after scaling. Writing

$I(Y;S)\le \epsilon,$4

the fairness constraint becomes $I(Y;S)\le \epsilon,$5, while feasibility also requires $I(Y;S)\le \epsilon,$6 and $I(Y;S)\le \epsilon,$7 (Zamani et al., 27 Nov 2025).

Using local information-geometric approximations, the paper shows that, for small $I(Y;S)\le \epsilon,$8, mutual informations can be approximated by quadratic forms. With

$I(Y;S)\le \epsilon,$9

the approximations are

$Y$00

This rewrites the fairness-design problem as a quadratic optimization problem $Y$01 over the perturbation directions $Y$02 and output probabilities $Y$03 (Zamani et al., 27 Nov 2025).

The spectral structure of the matrices yields closed-form characterizations in important cases. Proposition 3 states that both $Y$04 and $Y$05 have singular value $Y$06 with corresponding right singular vector $Y$07, and feasibility requires working in the orthogonal complement of that trivial direction. Theorem 2 then gives an explicit SVD-based construction: choose a binary representation $Y$08 with $Y$09, select perturbations along the principal feasible singular vector of $Y$10, and scale by a factor $Y$11 to satisfy the rate constraint. If the unscaled vector satisfies the compression budget, then $Y$12; otherwise the scaling is

$Y$13

For $Y$14, the resulting lower bound is tight for the approximated problem (Zamani et al., 27 Nov 2025).

A numerical example uses $Y$15 with

$Y$16

The paper computes

$Y$17

with singular values $Y$18 and $Y$19 for $Y$20, and $Y$21 and $Y$22 for $Y$23. For $Y$24 and $Y$25, the paper reports that the quadratic approximation $Y$26 is very close to the exact optimum $Y$27 in the high-fairness regime, while the mutual-information formulation $Y$28 dominates $Y$29, as expected from the stronger point-wise constraint (Zamani et al., 27 Nov 2025).

5. Base rates, Equalized Odds, and bounded disparity

In binary classification, statistical parity is usually expressed as

$Y$30

while Equalized Odds requires

$Y$31

equivalently

$Y$32

Base rates are

$Y$33

and the relation tying these quantities together is

$Y$34

This equation is the backbone of the compatibility analysis in the 2026 paper (Bargh et al., 26 Jan 2026).

Assuming Equalized Odds with common rates $Y$35 and $Y$36, statistical parity requires

$Y$37

The paper’s Theorem 1 therefore states that EO and SP are simultaneously satisfied either when base rates are balanced, $Y$38, or when $Y$39, which is the ROC chance line associated with a random classifier. Under unequal base rates, exact EO and exact SP are thus jointly feasible only at random-classifier operating points (Bargh et al., 26 Jan 2026).

The converse direction is geometrically symmetric. If SP is enforced with

$Y$40

then for each group

$Y$41

These are descending lines $Y$42 and $Y$43 in the $Y$44 plane. When $Y$45, the paper’s Theorem 2 states that the lines intersect at $Y$46, which lies on $Y$47. Again, the only exact EO-compatible point under base-rate imbalance is a random classifier (Bargh et al., 26 Jan 2026).

Although the paper notes that the term “bounded statistical parity” does not appear explicitly, it derives the quantitative structure needed for it. Under EO,

$Y$48

so

$Y$49

Using the paper’s notation, a natural bounded-SP definition is

$Y$50

In that case, achieving a given $Y$51 under EO requires

$Y$52

This makes the trade-off explicit: for fixed base-rate imbalance, tightening the SP bound forces the classifier toward the random line (Bargh et al., 26 Jan 2026).

The practical guidance in the paper follows directly from these equations. Before enforcing or relying on SP, practitioners should estimate base rates $Y$53; if they differ substantially, exact SP and exact EO together imply random guessing. The paper explicitly pleads for examining base-rate imbalance before enforcing or relying on the statistical-parity criterion (Bargh et al., 26 Jan 2026).

6. Interpretation, significance, and limitations

Across these works, bounded statistical parity functions simultaneously as a fairness guarantee and a privacy guarantee. In the mutual-information formulation, it bounds how many bits of information the representation reveals about the sensitive attribute. In the point-wise formulation, it ensures that for every released output, the posterior distribution of sensitive attributes remains close to the population prior. The point-wise paper states this dual interpretation directly: the constraint can be viewed both as a fairness guarantee and as a privacy guarantee (Zamani et al., 18 Aug 2025, Zamani et al., 27 Nov 2025).

The main distinction between the formulations is therefore between global and local control. A mutual-information constraint permits small average leakage, whereas a point-wise $Y$54 constraint rules out mechanisms in which average leakage is small but some outputs are highly revealing. This suggests a hierarchy: global bounded parity is more permissive, while local bounded parity provides a worst-case guarantee over outputs (Zamani et al., 27 Nov 2025).

The constructive methods also differ. The mutual-information work uses extended FRL and SFRL together with randomized mechanisms to derive upper and lower bounds and to show that non-zero leakage can improve utility (Zamani et al., 18 Aug 2025). The point-wise work uses local information geometry to reduce the design problem to a quadratic program and, in important cases, to a low-complexity SVD calculation (Zamani et al., 27 Nov 2025). A plausible implication is that the appropriate formalism depends on whether the design goal is average leakage control, per-output robustness, or explicit analysis of trade-offs with base rates and error parity.

Several limitations are stated explicitly. The mutual-information paper assumes discrete random variables $Y$55, although it notes that the results extend to continuous variables via continuous versions of SFRL. It also states that no explicit learning algorithm is provided; the work is information-theoretic and focuses on fundamental limits and constructions rather than neural-network training procedures (Zamani et al., 18 Aug 2025). The point-wise paper works in the small-$Y$56 regime, where local approximations of KL divergence and mutual information are valid, and its quadratic approximation is justified under constants depending on $Y$57 (Zamani et al., 27 Nov 2025). The EO-compatibility paper works with exact equalities and binary sensitive attributes, labels, and predictions, although its equations motivate approximate or bounded variants (Bargh et al., 26 Jan 2026).

Taken together, the literature treats bounded statistical parity not as a single metric but as a family of controlled relaxations of demographic parity. In one direction, the relaxation is global and information-theoretic, expressed by $Y$58. In another, it is local and point-wise, expressed by $Y$59 for every output. In classification analysis, it also appears as a bounded disparity whose feasible value is constrained by base-rate imbalance and Equalized Odds. The common theme is the replacement of exact independence by a quantitatively specified tolerance that can be analyzed in tandem with utility, compression, and error-rate structure (Zamani et al., 18 Aug 2025, Zamani et al., 27 Nov 2025, Bargh et al., 26 Jan 2026).