19-EFT.WP.Methods.SynthData v1.0 | Appendix E — Error & Uncertainty Propagation (Synthesis Edition)

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Appendix E — Error & Uncertainty Propagation (Synthesis Edition)

I. Scope & Objectives

• Provide a unified definition for the sources, combination, and publication of uncertainty across the synthetic-data lifecycle—design → training → generation → evaluation → release. Specify coverage reporting U = k * u_c, compatible with both offline batches and streaming increments.
• Deliverables: an uncertainty ledger err_budget.*, method tags method ∈ {analytic, bootstrap, posterior}, window & time-base fields, and mappings to contracts.* and manifest.synth.*.

II. Terms & Symbols

• Models & parameters: theta (generator parameters), hat_theta (estimate), Sigma_theta = Cov(hat_theta).
• Samples & weights: D_real, D_syn, n_real, n_syn, w_i, n_eff = ( ( ∑ w_i )^2 ) / ( ∑ w_i^2 ).
• Metrics & estimators: y = g(x) (metric functional), u(x) (standard uncertainty), u_c(y) (combined standard uncertainty), U = k * u_c (coverage uncertainty).
• Mechanism noise: sigma_dp (DP mechanism std or equivalent), eps_total, delta_total (privacy budget).
• Time base & arrival: tau_mono, ts, T_arr, delta_form, offset/skew/J.
• Uncertainty components: u_model (model estimation), u_sampling (finite sampling), u_dp (privacy), u_align (timebase/path), u_eval (evaluator/embedding), u_env (scenario/domain randomization).

*III. Axioms P40E- **

• P40E-1 (Explicit Measure): Declare domains and measures for all expectations, variances, and integrals; fix the embedding/preprocessing gauges and version them.
• P40E-2 (Coverage Publication): Publish u_c and U = k * u_c alongside every external metric, stating k and method provenance.
• P40E-3 (Traceable Components): The uncertainty ledger must separate {u_model, u_sampling, u_dp, u_align, u_eval, u_env}—no aggregation without lineage.
• P40E-4 (Time-Base Consistency): Compute windows on tau_mono, publish on ts, and record offset/skew/J plus both T_arr formulations with delta_form.
• P40E-5 (Dimensional Conservation): Run check_dim( y - f(x) ) before propagation; log any unit conversions explicitly.
• P40E-6 (Privacy-Noise Independence): Prefer the independence assumption between DP noise and data; if dependence exists, publish the covariance term or a valid upper bound.
• P40E-7 (Reproducibility): Publish the seeds/RNGs and bootstrap size B used to reproduce the uncertainty evaluation.

*IV. Minimal Equations S40E- **

• S40E-1 (Delta linearization): u_c^2( y ) = grad_x g(x)^T * Cov(x) * grad_x g(x)。
• S40E-2 (Parameter propagation): u_model^2( y ) = J_theta * Sigma_theta * J_theta^T，where J_theta = ∂g/∂theta |_{hat_theta}。
• S40E-3 (Finite sampling): u_sampling^2( y ) ≈ Var_hat( y | theta ) / n_eff。
• S40E-4 (Privacy-noise propagation): if y = g(x + n_dp) with n_dp ~ (0, Sigma_dp), then u_dp^2( y ) = J_x * Sigma_dp * J_x^T。
• S40E-5 (Arrival-time contribution):
  u_align^2( y ) = ( ∂g/∂T_arr )^2 * u^2( T_arr )，where
  u^2( T_arr ) = u_jitter^2 + ( delta_form^2 ) / 3 (uniform upper-bound approximation)。
• S40E-6 (Synthesis combination): if components are approximately independent,
  u_c^2 = u_model^2 + u_sampling^2 + u_dp^2 + u_align^2 + u_eval^2 + u_env^2;
  otherwise add 2 * ∑ Cov_i,j for correlated parts。
• S40E-7 (Bootstrap): u(y) = std( { y^(b) }_{b=1..B} )，CI = quantile( { y^(b) }, [alpha/2, 1-alpha/2] )。
• S40E-8 (Bayesian): U = k * sd( { g(theta^(s)) }_{s=1..S} )，or publish quantile band q_{alpha/2}, q_{1-alpha/2}。
• S40E-9 (Weighted SE): SE( mean_w )^2 = ( ∑ w_i^2 * (x_i - mean_w)^2 ) / ( ( ∑ w_i )^2 )。

V. Propagation Paths & Ledger Structure

1. Canonical chain (declare each u_*)
   • Design & calibration: hat_theta, Sigma_theta → u_model.
   • Sampling & generation: n_syn, w_i → u_sampling (including n_eff).
   • Constraints & alignment: enforce_constraints, align_timepath → u_align.
   • Privacy & watermark: DP mechanism + accounting → u_dp with {eps_total, delta_total}.
   • Evaluation & embeddings: FID/KID/MMD/W1/utility_gap → u_eval.
   • Domain randomization: scene-parameter variance → u_env.
   • Combine: u_c^2 = ∑ u_*^2 (+ correlations), publish U = k * u_c.
2. Ledger keys (suggested)
   • err_budget.model/sampling/dp/align/eval/env = {method, value, details}。
   • details must include at least B|S, kernel|backbone|layer, seed/rng, window, unit/dim.

VI. Synthesis-Specific Notes & Formulae

• Embedding metrics (e.g., FID)
  Approximation: u_eval^2( FID ) ≈ grad_{mu,Sigma} FID^T * Cov( mu,Sigma ) * grad_{mu,Sigma} FID；obtain Cov( mu,Sigma ) via asymptotics or bootstrap.
• Kernel-parameter sensitivity (e.g., MMD_RBF)
  u_eval^2 ≈ ( ∂MMD/∂h )^2 * u^2(h ) + a Delta-method sample term, where h is the bandwidth.
• DP synthesis (counts/histogram constraints)
  Gaussian mechanism: if c = true_count + n, n ~ N(0, sigma_dp^2), then u_dp^2 = sigma_dp^2; push through y=g(c) with J_c linearization.
• Conditional/controllable generation
  Condition uncertainty: u_env^2 = J_c * Cov(c) * J_c^T; if rejection sampling reduces n_eff, update u_sampling.
• Time-series / event synthesis
  Arrival-rate variance contributes to u_sampling; use block bootstrap for W1(inter_arrival) to avoid underestimating dependence.
• Multimodal consistency
  When aggregating across modalities, prefer median-of-means and publish u_agg; alternatively provide per-modality intervals and an upper bound on cross-metric correlations.

VII. Windows & Time-Base Alignment

1. Windowing policy
   Fixed span Delta_t with sliding step; require n_eff ≥ n_eff_min before publication; otherwise delay or widen the window.
2. Alignment requirements
   Record offset/skew/J; for path-involved gauges, publish both T_arr forms and delta_form with its u(T_arr).
3. Streaming recursion (mean & variance)
   • mu_{t+1} = mu_t + ( x_{t+1} - mu_t ) / n；
   • S_{t+1} = S_t + ( x_{t+1} - mu_t ) * ( x_{t+1} - mu_{t+1} )；
   • u = sqrt( S / ( n - 1 ) )；use n_eff for weighted variants.

*VIII. Contracts & Assertions C40E- **

• C40E-1 (Coverage Gating): assert( y + k * u_c(y) ≤ tol_y ) or require the two-sided band to cover the target interval.
• C40E-2 (Effective Sample Size): assert( n_eff ≥ n_eff_min ) (default recommendation n_eff_min = 128).
• C40E-3 (DP Budget): assert( eps_total ≤ eps_cap ∧ delta_total ≤ delta_cap ), recording the accounting method.
• C40E-4 (Arrival Consistency): assert( delta_form ≤ tol_Tarr ), add its contribution to u_align.
• C40E-5 (Method Disclosure): method ∈ {analytic, bootstrap, posterior} with all key parameters persisted; otherwise mark as non-compliant.
• C40E-6 (Dimensional Check): assert( check_dim( y - f(x) ) = pass ).

IX. I40- Implementation Bindings (Uncertainty)*

1. propagate_uncertainty_synth(report_in) -> err_budget
   • Inputs: hat_theta, Sigma_theta, metrics_raw, dp_config, align_info, env_cov.
   • Outputs: component u_*, u_c, U, and details.
2. bootstrap_metrics(ds_syn, metrics, B, seed) -> {u, CI, samples}
   Persist bootstrap resamples and intervals.
3. posterior_pushforward(posterior, g, S) -> {u, CI}
   Sample posterior theta^(s) and push through y=g(theta).
4. dp_accounting_and_variance(steps) -> {eps_total, delta_total, Sigma_dp}
   Return budgets and equivalent covariance from mechanism + accounting.
5. align_timepath_for_uncertainty(ds, sync_ref) -> {T_arr_form1, T_arr_form2, delta_form, u(T_arr)}
   Harmonize with I40-81 align_timepath and emit alignment uncertainty.
6. emit_uncertainty_manifest(err_budget) -> manifest.synth.metrics[*].u
   Write into manifest.synth.
7. Invariants: reproducible(seed); delta_form ≤ tol_Tarr; budgets eps_total, delta_total within limits; check_dim = pass; method and parameters recorded.

X. Cross-References

• Time-base & dual arrival forms: Methods.Cleaning v1.0 Chapters 6/10 and Appendices B/C.
• Imaging & embedding-metric uncertainty: Methods.Imaging v1.0 Chapter 14 and Appendices D/E.
• Statistical propagation & coverage conventions: Methods.CrossStats v1.0 Chapters 2/4/5 and Appendix E.

XI. Summary