19-EFT.WP.Methods.SynthData v1.0 | Chapter 11 — Bias, Fairness & Representativeness (Reweight / Mapping)

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Chapter 11 — Bias, Fairness & Representativeness (Reweight / Mapping)

I. Scope & Targets

1. Goals
   • Govern the representativeness of D_syn relative to p_target and improve group fairness via reweighting and distribution mapping (including optimal transport and monotone mappings), thereby reducing selection bias and coverage gaps.
   • Under intact physical/business constraints and privacy budgets, align both marginal and conditional distributions of the synthetic data with specified policy and compliance targets.
   • Publish auditable manifest.synth.fairness.* along with window-consistent operational metrics.
2. Applies to
   Tabular, time-series, graph, and multimodal data; offline and streaming synthesis; scenarios coupled to downstream training or evaluation.
3. Outputs
   A weight vector w(i) or a mapping T(x), plus an audit report, contract assertions, and a release manifest.

II. Terms & Variables

• Data & attributes: X (features), Y (label), A (protected attribute), C (condition/prompt), p_ref, p_syn, p_target.
• Reweighting: w(x), r(x) = p_target(x) / p_syn(x), w_clip, n_eff = ( (∑ w)^2 ) / ( ∑ w^2 ).
• Mapping & distances: T: X -> X', W1, MMD, KL, psi (population stability index).
• Fairness metrics: SPD = P(hat{Y}=1|A=1) - P(hat{Y}=1|A=0), EOD = (TPR_1-TPR_0, FPR_1-FPR_0), PPD (predictive parity diff).
• Time & arrival: tau_mono, ts, offset/skew/J, T_arr, delta_form.
• Manifest keys: manifest.synth.fairness.{method, targets, W1, MMD, psi, SPD, EOD, n_eff, windows, signature}.

*III. Axioms P411- **

• P411-1 (Explicit Targets): Before release, specify p_target (or a vector of group proportions) together with tolerances.
• P411-2 (Method Equivalence): Reweighting and mapping are equivalent at publication scope; provide impact analyses and rollback paths for both.
• P411-3 (Constraints First): Physical/business constraints and referential integrity (foreign keys) take priority over balancing actions.
• P411-4 (Unified Time Base): Evaluate representativeness and fairness on tau_mono; publish on ts with explicit windowing.
• P411-5 (Robustness): Any reweighting must ensure n_eff / N ≥ rho_min to prevent variance blow-up.
• P411-6 (No Privacy Degradation): Post-processing must not weaken DP(eps,delta); if retraining is required, re-account the budget.
• P411-7 (Dual Arrival Forms): When adjusting time/path fields, record both T_arr formulations and delta_form.
• P411-8 (Dimensional Conservation): Numerical mappings must pass check_dim(expr); no unit conflict may be introduced.
• P411-9 (Multimodal Consistency): Align multimodal bundles on the joint view; isolated per-modal balancing that breaks cross-modal coherence is disallowed.

*IV. Minimal Equations S411- **

1. S411-1 (Density-Ratio Reweighting)
   • r(x) = p_target(x) / p_syn(x)，w(x) = clip( r(x), 0, w_clip )，w_norm = w / ( (1/N) * ∑_i w_i )。
   • Effective sample size: n_eff = ( (∑_i w_i)^2 ) / ( ∑_i w_i^2 )。
2. S411-2 (Weighted Risk with Fairness Constraints)
   min_f E_{(x,y)~p_syn}[ w(x) * L( f(x), y ) ]
   s.t. | P_w( hat{Y}=1 | A=a ) - P_w( hat{Y}=1 | A=b ) | ≤ tol_spd，
   | TPR_w(a) - TPR_w(b) | ≤ tol_eod_TPR，| FPR_w(a) - FPR_w(b) | ≤ tol_eod_FPR。
3. S411-3 (Wasserstein Mapping)
   • T# p_syn = p_target，T = argmin_T E_{x~p_syn}[ c( x, T(x) ) ]，with c(x,z) = ||x - z||_2 commonly used。
   • Entropic OT: π* = argmin_π ⟨π, C⟩ + λ * H(π)，T(x) = ∑_j π*(x,x_j') * x_j'。
4. S411-4 (MMD Alignment)
   MMD^2 = || (1/N)∑ φ(x_i) - (1/M)∑ φ(x_j') ||_H^2 ≤ tol_mmd，declare kernel and bandwidth in the manifest。
5. S411-5 (Representativeness Ratio & Coverage)
   repr_ratio(a) = p_syn(A=a) / p_target(A=a)，covg = |supp(p_target) ∩ supp(p_syn)| / |supp(p_target)|。
6. S411-6 (Arrival-Time Consistency)
   T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )；T_arr = ( ∫ ( n_eff / c_ref ) d ell )；
   delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |。

V. Metrology Flow M40-11 (Representativeness & Fairness Loop)

1. Target Setting
   Define p_target (from a reference set or policy vector), protected attribute(s) A, windows, and thresholds.
2. Baseline Assessment
   Compute repr_ratio(a), W1, MMD, psi, and covg on tau_mono windows.
3. Method Selection
   If marginal proportion shifts dominate, begin with reweighting; for structural bias, use OT/monotone mapping or a hybrid.
4. Parameter Solving
   • Estimate r(x) (logistic-ratio | KLIEP | Bregman), obtain w_norm, and verify n_eff;
   • or solve for π* / T(x) while preserving foreign keys, units, and physical constraints.
5. Audit & Rollback
   Re-evaluate W1/MMD/psi/SPD/EOD; if any exceed thresholds, increase regularization, tighten w_clip, stratify alignment, or roll back to a coarser p_target.
6. Arrival-Time & Time-Base Handling
   Apply timepath_hardening; write offset/skew/J, T_arr, and delta_form.
7. Persist & Freeze
   Emit manifest.synth.fairness.* with signature; record methods, parameters, thresholds, and windowed results.

VI. Contracts & Assertions C40-11xx

• C40-1101 (Proportional Representativeness): For all a, |repr_ratio(a) - 1| ≤ tol_repr.
• C40-1102 (Structural Alignment): W1 ≤ tol_W1 and MMD ≤ tol_MMD and psi ≤ tol_psi.
• C40-1103 (Effective Sample Size): n_eff / N ≥ rho_min.
• C40-1104 (Fairness Constraints): |SPD| ≤ tol_spd, |TPR_1-TPR_0| ≤ tol_eod_TPR, |FPR_1-FPR_0| ≤ tol_eod_FPR.
• C40-1105 (Constraints & Units): assert_foreign_key = true and check_dim( T(x) - x ) valid.
• C40-1106 (Arrival Consistency): delta_form ≤ tol_Tarr; |offset| ≤ off_max, J ≤ J_max.
• C40-1107 (Multimodal Consistency): Joint-view alignment metrics must pass; single-modality pass is insufficient.
• C40-1108 (No Privacy Degradation): eps_total_after ≤ eps_total_before (post-processing does not weaken privacy guarantees).

VII. Implementation Bindings I40-11*

• estimate_density_ratio(ref, syn, method) -> r_hat
• compute_reweight(r_hat, clip, normalize) -> w_norm, n_eff
• fit_ot_map(syn, target, cost, reg) -> T, π*
• apply_mapping(ds_syn, T, constraints) -> ds_syn'
• audit_representativeness(ds_syn', ref, metrics) -> {W1, MMD, psi, repr_ratio, covg}
• audit_group_fairness(ds_syn', model_spec|metric) -> {SPD, EOD, PPD}
• timepath_hardening(ds_syn', sync_ref) -> ds_syn_t （writes T_arr, delta_form, offset/skew/J）
• emit_fairness_manifest(results, policy) -> manifest.synth.fairness
• Invariants: sum(w_norm)/N ≈ 1; n_eff increases monotonically with larger w_clip; foreign keys preserved; delta_form ≤ tol_Tarr; unit/dimension checks pass.

VIII. Cross-References

• Methods.CrossStats v1.0: Chapter 7 (drift & alignment), Chapter 9 (calibration transfer).
• This volume: Chapter 5 (robustness in deep generation), Chapter 9 (multimodal balancing), Chapter 12 (fidelity & utility evaluation), and Chapter 10 (privacy constraints).
• Methods.Cleaning v1.0: Chapter 10 (release freeze) and Appendix B (contract library).

IX. Quality SLIs & Risk Control

1. SLIs
   W1, MMD, psi, repr_ratio_p95, n_eff/N, |SPD|, |EOD|, latency_ms_p99 (alignment pipeline), delta_form.
2. Strategies
   • Divergent reweighting: tighten w_clip, stratify estimation, add regularization/smoothing.
   • Residual structural bias: switch to OT or segmented monotone mapping; coarsen p_target if needed.
   • Fairness–utility conflict: multi-objective trade-offs or relaxed constraints; cost-sensitive learning with stability validation.
   • Streaming drift: update w/T on sliding windows and integrate with alert/rollback.

Summary