35-EFT.WP.EDX.OrientedTension v1.0 | Chapter 9 Data & Uncertainty: Inversion, Constraints & Propagation

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Chapter 9 Data & Uncertainty: Inversion, Constraints & Propagation

I. Abstract & Scope
This chapter defines an integrated workflow for oriented-tension problems that joins data → parameters → derived quantities: a unified DatasetBundle, registered likelihood families and priors, posterior computation with evidence, and uncertainty propagation—via linearization and sampling—to constitutive/coupling parameters and the energy ledger (EDX), harmonized with dominance masks. All symbols use English notation in backticks; SI units apply. If ToA fields are present, both forms must be recorded in parallel with explicit {gamma(ell), d ell}.

II. Dependencies & References

• Geometry & orientation: Chapter 3 S80-1/2.
• Axioms & minimal equations: Chapter 4 S80-3/4.
• Metrology & calibration: Chapter 5 M80-1…4.
• Couplings & media: Chapter 6 S80-5/6.
• Energy accounting: Chapter 7 S80-7/8.
• Numerics & implementation: Chapter 10 (SimStack-OT), Chapter 12 (I80-*).

III. Normative Anchors (added in this chapter, S80-/M80-)

• M80-25 (DatasetBundle & prior registry): merge multi-source data, covariances, units/dimensions, and see: anchors into a DatasetBundle; register parameter priors and physical constraints.
• M80-26 (Likelihood families & noise models): unified interface for Gaussian/Poisson/mixed likelihoods and instrument kernel R_inst.
• M80-27 (Posterior computation & evidence): obtain {posterior, Z, logZ} and information criteria via nested sampling/SMC.
• M80-28 (Uncertainty propagation): propagate by linearization (Fisher/Delta) and by sampling to {Λ_{ijkl}, A, K, tau_relax, D_Q, χ_*, D1, α_*, κ_s} and to {𝒫_*, Φ_E, W_orient}.
• M80-29 (Physical constraints & feasible domain): enforce positivity, objectivity, symmetry, and nonnegative energy via hard/soft constraints.
• M80-30 (Dominance-mask harmonization): incorporate Chapter 6 η_dom masks in likelihoods/predictions to prevent cross-channel leakage fits.
• S80-12 (Posterior & evidence): p(θ|D) ∝ L(D|θ) π(θ), Z = ∫ L(D|θ) π(θ) dθ.
• S80-13 (Linearized propagation): Cov_g ≈ J_g Cov_θ J_g^T, J_g = ∂g/∂θ.
• S80-14 (Fisher information): F = E[ − ∂^2 log L / ∂θ∂θ^T ], with Cov_θ ≈ (F + Π)^{-1} (Π prior precision).

IV. Body Structure

I. Background & Problem Statement

• Oriented-system data are multimodal, multiscale, and noise-heterogeneous. A unified DatasetBundle with likelihood families must couple the constitutive/coupling/energy models so that evidence is comparable and uncertainties are physically meaningful under hard constraints.
• Objective: from Q_ij, T_fil_ij, and multiphysics observations, obtain posteriors usable for engineering/scientific prediction with credible intervals on derived quantities.

II. Key Equations & Derivations (S-series)

1. S80-12 (Posterior & evidence): p(θ|D) ∝ L(D|θ) π(θ), Z = ∫ L π dθ; evidence ratio K = Z_1/Z_0 supports model comparisons (with/without coupling, isotropic/anisotropic, etc.).
2. Likelihood exemplars:
   • Gaussian: log L_G = − (1/2) (y − 𝒦[θ])^T Σ^{-1} (y − 𝒦[θ]) + const.
   • Poisson: log L_P = ∑_i ( k_i log λ_i(θ) − λ_i(θ) − log k_i! ).
   • Mixed/correlated noise: via covariance kernels or spectral PSDs.
3. S80-13 (Delta/Fisher propagation): Cov_g ≈ J_g (F + Π)^{-1} J_g^T, for g(θ) ∈ {𝒫_*, Φ_E, W_orient, c(ê), Δn(ê)}.
4. S80-14 (Predictive distribution): p(y_*|D) = ∫ p(y_*|θ) p(θ|D) dθ for banded ledgers and masked predictions.

III. Methods & Flows (M-series)

1. M80-25 DatasetBundle & Priors
   • Collect {polarimetry, transport, waves, mechanics} with UnitsAudit.log.
   • Register priors from ModelCard/ParameterCard within physical feasibility.
   • Integrate R_inst, covariances, and see: anchors to form DatasetBundle.
2. M80-26 Likelihoods & Noise
   • Choose L_G/L_P/L_mix with correlated-noise kernels as needed.
   • Fold deconvolution/regularization residual spectra from Chapter 5 into noise estimates.
   • Apply dominance-mask weights across energy/frequency bands.
3. M80-27 Posterior & Evidence
   • Run nested sampling/SMC to obtain {posterior, Z, logZ}.
   • Output marginals, correlation matrices, and convergence diagnostics.
   • Produce evidence comparisons for {with/without coupling, isotropic/anisotropic}.
4. M80-28 Uncertainty Propagation
   • Linearization: compute J_g and Cov_g.
   • Sampling: push posterior draws through the forward map 𝒦 to {𝒫_*, Φ_E, W_orient, c(ê), Δn(ê)}.
   • Produce energy ledgers and directional/banded confidence envelopes.
5. M80-29 Physical-Constraint Enforcement
   • Soft: add penalties in the objective (positivity/objectivity/symmetry).
   • Hard: reparameterize (e.g., D_eff = L L^T).
   • A posteriori: discard infeasible samples.
6. M80-30 Mask Harmonization & Prediction
   • Embed η_dom masks in likelihoods and predictive averaging.
   • Generate segmented predictions/uncertainties without cross-channel leakage.
   • Output partitions consistent with Chapter 7 ledgers.

IV. Cross-References within/beyond this Volume

• Chapter 4: harmonize constitutive/dynamic parameter posteriors and uncertainties (S80-3/4).
• Chapter 5: use metrology posteriors as priors/likelihood inputs and calibrate noise.
• Chapter 6: coordinate coupling parameters and dominance masks in inversion/prediction.
• Chapter 7: decompose uncertainties of power terms and ledgers; closure audits.
• Chapters 10/12: implement samplers and forward operators in SimStack-OT and I80-*.

V. Validation, Criteria & Counterexamples

1. Positive criteria:
   • Models with coupling/anisotropy show significantly higher logZ than baselines and reduce ledger-closure residuals.
   • Predictive intervals achieve nominal coverage on independent data.
   • Physical constraints (positivity, objectivity, symmetry) are satisfied with no systematic residual bias.
2. Negative criteria:
   • Removing key couplings or collapsing Q_ij to isotropy does not reduce evidence.
   • Large, unexplained discrepancies between Fisher-based and sampling propagation.
   • Banded ledgers inconsistent with global ledgers beyond CIs.
3. Contrasts:
   • Evidence/prediction differences among {Gaussian, Poisson, Mixed} noise models.
   • Confidence-band comparisons {linearization vs sampling}.
   • {with mask vs without mask} impacts on leakage.

VI. Deliverables & Figure List

1. Deliverables:
   • DatasetBundle.tar (data, covariances, R_inst, units/dimensions).
   • Posterior.zarr (parameter posteriors & correlations), Evidence.txt.
   • Predictive.nc (predictive bands for derived quantities & ledgers).
   • ConstraintReport.md (physical-constraint and feasibility checks).
2. Figures/Tables (suggested):
   • Tab. 9-1 Likelihood & prior registry checklist.
   • Fig. 9-1 Posterior marginals and correlation heatmaps.
   • Tab. 9-2 Consistency of Fisher vs sampling propagation.
   • Fig. 9-2 Closure comparison: banded vs global ledgers.
   • Tab. 9-3 Evidence comparisons and positive/negative criteria summary.