18-EFT.WP.Methods.CrossStats v1.0 | Chapter 2 — Axioms and Minimal Equations (Statistical Baseline)

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Chapter 2 — Axioms and Minimal Equations (Statistical Baseline)

One-Line Objective

I. Scope and Objects

1. Scope
   • Applies to descriptive statistics, parameter estimation, intervals/posteriors, robust variance, sequential decisions, and arrival-time–aware windowed statistics for both batch data and event streams.
   • Compute on tau_mono, publish on ts; whenever T_arr is involved, compute both conventions in parallel and record delta_form.
2. Objects
   Data D = { (x_i, y_i, t_i, w_i, m_i) }, model parameters theta, prior p(theta), likelihood L(theta; D), statistical window Delta_t, and contract policy policy.

II. Terms and Variables

• Sampling and weights: pi(i) (inclusion probability), w_i = 1 / pi(i), W_norm = ( ∑ w_i ) / N, n_eff = ( ∑ w_i )^2 / ( ∑ w_i^2 ).
• Estimation and uncertainty: hat{theta}, SE, CI, U = k * u_c.
• Timebase and time of arrival: tau_mono, ts, offset/skew/J; two T_arr conventions:
  T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ), T_arr = ( ∫ ( n_eff / c_ref ) d ell ), discrepancy delta_form.
• Information and diagnostics: I(theta) (Fisher information), H (Hessian), s_i (per-observation score), ppc (posterior predictive check).

III. Axioms P302-*

• P302-1 (Exchangeability / hierarchical exchangeability): within a design stratum, treat samples as exchangeable; across strata, use weights or hierarchical models.
• P302-2 (Consistent sampling frame & explicit weights): any weighted analysis must document how pi(i) was generated and versioned, and must retain weight metadata and the normalization factor.
• P302-3 (Dimensional conservation): before cross-modal aggregation, run check_dim(expr); unit-repair failures constitute contract violations.
• P302-4 (Unified timebase): evaluate windows on tau_mono; publish on ts with offset/skew/J.
• P302-5 (Parallel arrival-time conventions): for any statistic involving arrival time or propagation delay, compute both conventions and assert delta_form ≤ tol_Tarr.
• P302-6 (Explicit and interpretable priors): Bayesian analyses must state the motivation for p(theta) and include sensitivity analysis.
• P302-7 (Robustness and diagnostics): provide sandwich (robust) variance and posterior predictive checks by default; account for sequential/multiple-comparison alpha consumption in a unified ledger.
• P302-8 (Declared missingness mechanism): encode missingness via m_i ∈ {0,1}; explicitly document the assumed mechanism (MCAR/MAR/MNAR) and imputation strategy; implicit fills are prohibited.

IV. Minimal Equations S302-*

• S302-1 (Weighted log-likelihood and score):
  ell(theta) = ∑_i w_i * log p( y_i | x_i; theta );
  U(theta) = ∂ ell(theta) / ∂ theta = ∑_i w_i * ∂ log p( y_i | x_i; theta ) / ∂ theta;
  hat{theta} satisfies U(hat{theta}) = 0.
• S302-2 (Fisher information and asymptotic normality):
  I(theta) = - E[ ∂^2 ell(theta) / ∂ theta ∂ theta^T ];
  hat{theta} ~ approx Normal( theta, I(theta)^{-1} ) (with design- or replicate-based correction under weighting).
• S302-3 (Sandwich robust covariance):
  H = - ∑_i w_i * ∂^2 log p_i / ∂ theta ∂ theta^T;
  S = ∑_i w_i^2 * s_i * s_i^T, where s_i = ∂ log p_i / ∂ theta;
  Var_robust( hat{theta} ) = H^{-1} * S * ( H^{-1} )^T.
• S302-4 (GLM score equations, unified notation):
  g( mu_i ) = x_i^T beta, mu_i = E[ Y_i | x_i ];
  U(beta) = X^T W ( y - mu ) = 0 (matrix W incorporates w_i, the variance function, and the link-derivative factor).
• S302-5 (Bayesian posterior and posterior predictive):
  p(theta | D) = k * L(theta; D) * p(theta), with normalizing constant k;
  p( y_new | x_new, D ) = ( ∫ p( y_new | x_new, theta ) * p( theta | D ) d theta ).
• S302-6 (Delta method):
  Var( g( hat{theta} ) ) ≈ ( ∇g( theta ) )^T Var( hat{theta} ) ( ∇g( theta ) ).
• S302-7 (Intervals and coverage):
  Frequentist: CI_{1-α} = hat{theta} ± z_{1-α/2} * SE( hat{theta} );
  Bayesian: CI_{1-α} = [ q_{α/2}( p(theta|D) ), q_{1-α/2}( p(theta|D) ) ].
• S302-8 (Windowed statistics):
  hat{mu}_w( t; Delta_t ) = ( ∑_{i: |tau_i - t| ≤ Delta_t/2} w_i y_i ) / ( ∑_{i: |tau_i - t| ≤ Delta_t/2} w_i ).
• S302-9 (Two-sample effects and sample-size approximation):
  Mean difference delta = mu_A - mu_B, with variance Var( delta ) = sigma_A^2 / n_A + sigma_B^2 / n_B;
  for equal variances and equal per-arm sample sizes,
  n_per_arm ≈ ( 2 * ( z_{1-α/2} + z_{power} )^2 * sigma^2 ) / delta^2.
• S302-10 (Sequential GLR statistic and stopping):
  Lambda_t = ( sup_{theta ∈ H1} L_t( theta ) ) / ( sup_{theta ∈ H0} L_t( theta ) );
  tau = inf { t : log Lambda_t ≥ h_1 or log Lambda_t ≤ - h_0 }.
• S302-11 (Foundational drift metrics):
  KL( P_ref || P_cur ) = ( ∫ p_ref * log( p_ref / p_cur ) dx );
  W1( P_ref, P_cur ) = ( ∫ | F_ref(x) - F_cur(x) | dx );
  psi = ( ∑ bins ( p_cur - p_ref ) * log( p_cur / p_ref ) ) (discrete approximation).

V. Statistical Process M30-2 (Baseline → Diagnostics → Release)

• Readiness
  Verify units and dimensions; align to tau_mono; load the weighting scheme and normalize with W_norm ≈ 1; define H0/H1 and alpha/power, or the prior p(theta).
• Estimation
  Solve U( hat{theta} ) = 0 or draw posterior samples; compute SE, Var_robust, or posterior quantiles; evaluate windowed metrics over Delta_t.
• Diagnostics
  Inspect H condition number and near-zero scores; compare robust vs. model-based variance; run ppc and convergence checks; record alpha consumption for sequential or multiple comparisons.
• Release
  Produce CI/posterior, decisions, and logs; record both T_arr conventions and delta_form in parallel; persist manifest.stats.* and sign for freeze.

VI. Contracts and Assertions

• C30-21 (Weighting convention): | W_norm - 1 | ≤ tol_w_norm; max(w_i) / median(w_i) ≤ cap_w_max.
• C30-22 (Hessian and identifiability): H positive definite with cond(H) ≤ cap_cond; || U( hat{theta} ) || ≤ tol_score.
• C30-23 (Coverage/power): offline playback coverage |cov - target_cov| ≤ tol_cov; online alpha_spent ≤ alpha_budget.
• C30-24 (Bayesian diagnostics): Rhat ≤ cap_rhat, ESS/N ≥ min_ess_ratio, ppc_fail_rate ≤ tol_ppc.
• C30-25 (Arrival-time and windows): delta_form ≤ tol_Tarr; window coverage cov_rate ≥ tol_window_cover.

VII. Implementation Bindings I30-*

• I30-41 fit_glm(ds, formula, family) -> model: returns hat{theta}, SE, Var_robust, diagnostics, and residuals; invariant cond(H) ≤ cap_cond.
• I30-42 fit_bayes(ds, model_spec, priors) -> posterior: returns samples, summaries, ppc, and diagnostics; invariant Rhat ≤ cap_rhat.
• I30-43 bootstrap_metric(fn, ds, B) -> {est, CI}: BCa or percentile; invariant B ≥ B_min.
• I30-44 sequential_glrt(stream, H0, H1, h) -> stop_time: returns tau, decision, and alpha_spent.
• I30-45 emit_stats_manifest(results, policy) -> manifest.stats: writes contract outcomes, thresholds, TraceID, and signature.

VIII. Cross-References

• Schema binding and unit repair: see Methods.Cleaning v1.0, Ch. 3/4.
• Timeline and arrival time: see Methods.Cleaning v1.0, Ch. 5/6.
• Drift monitoring and alerts: see this volume, Ch. 7; experimental design: Ch. 8; causal inference: Ch. 10.
• Imaging quality mapping and metrics: see Methods.Imaging v1.0, Ch. 14.

IX. Quality and Risk Control

• SLI/SLO
  latency_ms_p99 for estimation, coverage deviation, robust-to-model variance ratio, alpha_spent/alpha_budget, and Rhat shortfall rate.
• Rollback
  Contract failures trigger rollback to the previous signed manifest.stats; retain TraceID and the runtime environment digest.

Summary