14-EFT.WP.Methods.Inference v1.0 | Chapter 3: Principles of Inference & Postulates

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Chapter 3: Principles of Inference & Postulates

I. Scope & Objectives

1. Establish the foundational postulates P41-* and minimal equations S42-* for the inference domain, covering risk minimization, calibration consistency, time-base alignment, and offline/online equivalence. These serve as verifiable grounds for Chapter 6 (offline/online parity), Chapter 7 (calibration & uncertainty), and Chapter 12 (acceptance scoring).
2. Target outputs
   • Citable anchors for postulates and equations.
   • Metrology flows Mx-4* aligned with EnvLock, Graph(theta), and TS.*.
   • Mathematical definition of the unified acceptance gate gate.inf and suggested threshold fields.
3. Pass criteria
   • Any expression containing division or an integral is enclosed in parentheses and declares the path gamma(ell) and measure d ell.
   • Offline/online comparisons are first mapped to the common time base ts = alpha + beta * tau_mono.
   • Calibration metrics must disclose bin boundaries, weighting, and sample counts.

II. Terms & Symbols

• Probability & risk: p(y | x, theta), p(theta | D), y_hat, L(y, y_hat), R_emp, R_exp, NLL.
• Calibration & uncertainty: ECE, MCE, mean, var, quantile(q), interval(1 - delta).
• Consistency: delta_offon = ( norm( y_hat_off - y_hat_on ) / norm( y_hat_off ) ), R_infer = 1 - delta_offon.
• Time base: tau_mono, ts, alpha, beta, window = [t0, t1].
• Spectrum & energy: S_xx(f), U_w, ENBW, var( x ) ≈ ( ∫ S_xx(f) df ).
• Environment & bindings: EnvLock, Graph(theta), hash(•), fingerprint, anchor, TS.latency/TS.thrpt/TS.error.

III. Postulates & Minimal Equations

1. P41-1 Equivalence of inference (fixed graph & environment)
   With fixed Graph(theta) and EnvLock, given the same anchor, identical seed, and nondet_guard = true, for any input x:
   • run_inference_off(x) = run_inference_on(x).
     If nondet_guard = false while rng_family/seed remain identical, then the distributions agree:
   • p_off( y_hat | x ) = p_on( y_hat | x ).
2. P41-2 Time-base alignment postulate
   Offline replay and online real-time must satisfy ts = alpha + beta * tau_mono; any cross-device/domain comparison must first map to the common time base, then perform window alignment and scoring.
3. P41-3 Monotone, mass-preserving calibration
   Σ_k w_k * g_bar_k = mean( p ), with w_k the in-bucket weights.The calibration map g(p) is monotone and preserves event ordering: if p_i >= p_j then g(p_i) >= g(p_j). For bucketed estimates, total probability mass is conserved:
4. S42-1 Risk minimization & bias–variance (squared-loss form)
   Empirical risk: R_emp = ( 1 / N ) * Σ_{i=1..N} L( y_i, y_hat_i ).
   Regularized ERM: theta_star = argmin_theta ( R_emp + lambda * Omega(theta) ).
   Error decomposition: E[ ( y_hat - y )^2 ] = ( bias )^2 + variance + sigma_eps2, where sigma_eps2 is irreducible noise.
5. S42-2 Minimal equations for probability calibration (ECE/MCE/NLL)
   ECE = Σ_{k=1..K} w_k * | acc_k - conf_k |, MCE = max_k | acc_k - conf_k |;
   NLL = - ( 1 / N ) * Σ log p_theta( y_i | x_i ).
   Bucketing must declare boundaries {b_0,...,b_K} and w_k = n_k / N.
6. S42-3 Offline/online consistency
   delta_offon = ( norm( y_hat_off - y_hat_on ) / norm( y_hat_off ) ), R_infer = 1 - delta_offon; example gate: delta_offon <= tau_offon.
   For time-series tasks, additionally report spectral parity:
   delta_psd = ( ∫ | S_xx_off(f) - S_xx_on(f) | df ) / ( ∫ S_xx_off(f) df ), with window U_w and ENBW specified.
7. S42-4 Two arrival-time formulations & path declaration (inherited cross-volume)
   Compute in parallel: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ) and T_arr = ( ∫ ( n_eff / c_ref ) d ell ); report delta_form and specify gamma(ell) with d ell.

IV. Data & Manifest Conventions

• Training/evaluation distribution statement: p_train(x,y), p_eval(x,y). If a domain shift exists, provide the divergence measure shift = D( p_train || p_eval ) and its rationale.
• Minimal calibration record: {id, ts, y, y_hat, conf, bin_index, weight}; bucket config {K, {b_k}} must be ingested and versioned.
• Consistency comparison manifest
  Matching key match_key = {id or (ts, source, window)}; alignment parameters {alpha, beta}; window_tol; the definition and units of norm(•).

V. Algorithms & Implementation Bindings

1. Bound prototypes (excerpt)
   • I40-4 score_predictions(y_true:any, y_pred:any, metrics:dict) -> ScoreReport
   • I40-5 calibrate(runtime:Runtime, method:str, data:any) -> CalibReport
   • I40-7 monitor_drift(stream:any, spec:dict) -> DriftReport
   • I40-10 compare_offline_online(off:any, on:any, policy:dict) -> ConsistencyReport
2. Sketch flows
   • Risk minimization: theta_star ← argmin ( R_emp + lambda*Omega ); emit BiasVarReport.
   • Probability calibration: q ← g(p), optimize ECE or NLL; produce CalibReport and parameters of g.
   • Offline/online parity: after ts alignment, compute delta_offon and delta_psd; generate ConsistencyReport.

VI. Metrology Flows & Run Diagram

• Mx-41 Build & lock: freeze EnvLock, load Graph(theta), finalize ParamCard and alpha,beta.
• Mx-42 Risk & calibration: evaluate R_emp/R_exp, perform bias–variance analysis, run calibration and output CalibReport with ECE/NLL.
• Mx-43 Offline/online parity: replay production windows, compute delta_offon and delta_psd; if thresholds are exceeded, trigger rollback playbooks.
• Mx-44 Gate synthesis & release: consolidate acc/ECE/NLL/TS.*/delta_offon, evaluate gate.inf, and produce the acceptance decision.

VII. Verification & Test Matrix

1. Minimum cases
   • Determinism: nondet_guard = true; rerun with the same seed, assert identical y_hat (tests P41-1).
   • Time-base alignment: sample random windows; after verifying ts = alpha + beta * tau_mono, assert delta_offon within gates (tests P41-2).
   • Calibration robustness: compare ECE under multiple K-bucket schemes (tests P41-3 and S42-2).
   • Spectral parity: for sequence tasks, verify var( x ) ≈ ( ∫ S_xx(f) df ) and delta_psd within gates (tests S42-2 extension).
   • Quantization switch: fp32 → int8 equivalence; record accuracy_drop and delta_offon.
2. Boundary & extreme
   Very low-confidence samples, long-tail classes, missing features, dynamic batch sizes, device thermal throttling elevating TS.latency_p95.

VIII. Cross-References & Dependencies

• With Core.Metrology: spectral convention S_xx(f), windows U_w/ENBW, and aggregation of error metrics.
• With Core.Threads: concurrency semantics and TS.*; rate limiting and batching conventions for their impact on numerical inference paths.
• With EFT.WP.Methods.Repro Chapter 3: mapping between reproducibility delta_rep and this chapter’s delta_offon; inheritance of environment lock and signatures.

IX. Risks, Limitations & Open Questions

• Incomplete masking of nondeterminism: parallel reductions, GPU atomic ordering, library-internal randomness; record rng_family/seed and enforce guardrails prior to gate.inf.
• Calibration extrapolation risk: behavior of g(p) outside support is not guaranteed; recommend conservative clipping p ∈ [eps, 1 - eps].
• Domain/Concept drift: p_train ≠ p_eval can inflate ECE or distort NLL; combine drift reports with re-calibration.
• Nonlinear amplification under precision switches: for some operator families, delta_offon vs. accuracy_drop can be non-monotonic; require operator-level regression.

X. Deliverables & Versioning

1. Deliverables
   • Postulates.md with the finalized P41-*, S42-* text and change fingerprints.
   • CalibReport and ConsistencyReport (including {K,{b_k},ECE,MCE,NLL,delta_offon,delta_psd}).
   • BiasVarReport (with bias^2/variance/sigma_eps2).
   • gate.inf configuration (tau_offon,tau_ece,tau_lat,...) and signature fingerprint.
2. Versioning policy
   • Any change to Graph(theta), EnvLock, K/{b_k}, or alpha,beta bumps the minor version and triggers full Mx-43 regression.
   • Report/visual-only updates do not trigger re-calibration but must roll the fingerprint and append change records (see Appendix C).