03-EFT.WP.Core.Parameters v1.0 | Chapter 7 — Uncertainty and Propagation

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Chapter 7 — Uncertainty and Propagation

I. Aims and Scope

• Establish a unified mapping from parameter uncertainty to output uncertainty, covering two families of methods: linearization (Delta) and sampling (MC/QMC), and report mean/std/Cov/Corr/CI/VaR/CVaR and related metrics.
• Provide the dedicated propagation formula for arrival time T_arr = ( ∫ ( n_eff / c_ref ) d ell ), with the path gamma(ell) and the measure d ell stated explicitly.
• Bind to implementation interfaces I30 7/8/10 and workflow Mx-3, maintaining dimensional and symbolic consistency with Identifiability (Chapter 5) and Calibration (Chapter 6).

II. Uncertainty Classes and Notation

1. Sources
   • Epistemic (parameters): theta ~ post(theta | data) or the prior prior(theta).
   • Aleatoric (observation/process noise): y = f(x; theta) + ε, ε ~ N(0, Σ) or other families.
2. Targets
   • Scalar functionals: g(theta) (e.g., T_arr(gamma; theta)).
   • Field/vector outputs: y = f(x; theta), evaluated on domain Ω or a discrete grid.
3. Statistics
   E[•], Var[•], Cov[•], Corr[•], CI_{1-α}[•], VaR_α[•], CVaR_α[•].

III. Linearized Propagation (Delta Method, Minimal Equation S71-1)

1. For a scalar output z = g(theta), first-order about theta_hat:
   Var[z] approx grad_theta[g]^T * Cov[theta] * grad_theta[g]
2. For a vector output y = f(theta) ∈ R^m:
   • Cov[y] approx J_f * Cov[theta] * J_f^T
   • where J_f = ∂ f / ∂ theta |_{theta=theta_hat} (see Chapter 5 S51-1 and interface compute_jacobian).
3. Validity
   f is approximately linear in a neighborhood of theta_hat; Cov[theta] is bounded and estimable (from Chapter 6 MCMC or a Fisher approximation).

IV. Uncertainty Propagation for Arrival Time (Minimal Equation S71-2)

1. Continuous form
   • g(theta) = T_arr( gamma; theta ) = ( ∫_{gamma} ( n_eff(x; theta) / c_ref(theta) ) d ell )
   • ∂ g / ∂ theta_i = ( ∫_{gamma} ( ( ∂ n_eff / ∂ theta_i ) / c_ref - n_eff * ( ∂ c_ref / ∂ theta_i ) / ( c_ref^2 ) ) d ell )
   • Var[g] approx grad_theta[g]^T * Cov[theta] * grad_theta[g]
2. Discrete realization (with discretize_path, propagate_time)
   • T_arr approx Σ_{j=1..N_γ} ( n_eff(x_j; theta) / c_ref(theta) ) * Δ ell_j
   • ∂ g / ∂ theta_i approx Σ_{j} ( ( ∂ n_eff(x_j) / ∂ theta_i ) / c_ref - n_eff(x_j) * ( ∂ c_ref / ∂ theta_i ) / ( c_ref^2 ) ) * Δ ell_j
3. Dimensional check
   The integrand ( n_eff / c_ref ) * d ell is dimensionless; admit only after check_dim(expr) passes.

V. Nonlinear Propagation: Monte Carlo and Quasi–Monte Carlo (Minimal Equation S71-3)

1. Sampling schemes
   • Posterior propagation: theta^{(s)} ~ post(theta | data); prior propagation: theta^{(s)} ~ prior(theta).
   • Sampling families: MC, QMC (Sobol), LHS; prefer QMC/LHS to reduce variance.
2. Estimators
   • μ_y approx ( 1 / S ) * Σ_{s=1..S} f(theta^{(s)})
   • Cov[y] approx covariance( { f(theta^{(s)}) } )
   • CI_{1-α}[y] via quantiles or normal approximation.
3. Convergence and budget
   • Monitor standard error SE(μ_y) and stabilized quantiles; stop when max_j |Δ q_j| < ε_q for K consecutive checks.
   • Under cost constraints, prioritize sample coverage S over grid refinement.

VI. Predictive Uncertainty and Intervals (Minimal Equation S71-4)

1. Predictive distribution
   • p(y_new | data) = ∫ p(y_new | theta) * post(theta | data) d theta
   • Approximation: draw { y_new^{(s)} ~ p(• | theta^{(s)}) }, then form CI_{1-α}[y_new] and PI_{1-α}[y_new].
2. Risk metrics
   • VaR_α[y] = quantile_α(y); CVaR_α[y] = E[ y | y ≥ VaR_α[y] ]
   • Useful for exceedance risks on arrival time, energy budgets, etc.

VII. Uncertainty Budget and Contributions (Minimal Equation S71-5)

1. Linearized variance decomposition (by parameter)
   • Var[z] approx Σ_i Σ_j g_i * Cov[theta]_{ij} * g_j, where g_i = ∂ g / ∂ theta_i
   • Group contribution for theta_g: Var_g[z] = g_g^T * Cov[theta_g] * g_g + cross_terms
2. Sobol indices (global)
   • S_i = Var( E[ z | theta_i ] ) / Var(z); S_{Ti} is the total effect.
   • Implementation note: share the sample bank between global_sensitivity_sobol and propagate_uncertainty_mc for reuse.

VIII. Consistency for Windowing and Coarse-Graining

• Time average: bar_y_t = avg_t[y; Δt]
  If y is stationary with integral time τ_int, then Var[ bar_y_t ] approx Var[y] * ( τ_int / Δt )
• Volume average: bar_y_V = avg_V[y; V=Ω]
  Var[ bar_y_V ] depends on correlation volume; estimate via block re-sampling or a dual spatial grid.
• Path average: bar_y_γ = avg_gamma[y]
  Tied to discrete path nodes; must share Δ ell_j with discretize_path.

IX. Non-Dimensionalization, Transforms, and Inverse Mapping

• Covariance in transform space
  With phi = T_map(theta), theta = T_map^{-1}(phi),
  Cov[theta] approx G * Cov[phi] * G^T, G = ∂ T_map^{-1} / ∂ phi |_{phi_hat}
• Output scaling
  Report Cov[bar_y] using bar_y = y / y0, then restore to physical units as needed.
• When MAP/MCMC is performed in phi-space, propagation back to the physical theta space must include the Jacobian term (see Chapter 3).

X. Uncertainty Propagation Pipeline Mx-3 (Standard Steps)

• State objectives and windows: declare the target g(•) and the windows/domains for avg_t/avg_V/avg_gamma.
• Converged parameter law: obtain Cov[theta] (Fisher or posterior samples) and Corr[theta].
• Fast linearized pass: use compute_jacobian to get J_f, compute Cov[y] approx J_f Cov[theta] J_f^T.
• Choose sampling strategy: select MC/QMC/LHS by nonlinearity strength and constraints; set the sample budget.
• Generate samples: posterior_sample_mcmc or sample_prior, map back via inverse_transform to the physical domain, apply lb/ub.
• Forward propagate: batch-call the model or propagate_time (for arrival time) to produce { f(theta^{(s)}) }.
• Summarize & diagnose: compute mean/Cov/CI/VaR/CVaR; check convergence and extremal stability.
• Budget & contributions: produce parameter/group contribution tables and Sobol indices.
• Report & versioning: export_params("yaml") and scenario snapshots; record random seeds and sampling family.
• Regression baseline: compare with historical uncertainty curves via compare_param_sets.

XI. Implementation Binding and Interface Mapping

• Statistics & covariance: covariance(samples), correlation(samples), regularize_cov(Cov, "shrinkage", alpha)
• Propagation: propagate_uncertainty_mc(model, prior_spec, n) (or substitute posterior samples for prior_spec)
• Global sensitivity: global_sensitivity_sobol(model, ranges, outputs, n)
• Cross-volume gradients: compute_jacobian(eqn:IRef, params:list[str]) for the linearization in S71-1/2
• Dimensional check: check_dim(expr); path discretization: discretize_path; arrival time: propagate_time

XII. Misuse and Conflict Checklist

• Mixing n and n_eff; or writing arrival time as ∫ n d ell / c (violates parentheses and canonical form; see global settings and Chapter 6).
• Omitting the path declaration gamma(ell) and d ell; or making Δ ell_j inconsistent with the reported window.
• Using linearization alone in strongly nonlinear regimes; or replacing a non-Gaussian posterior’s true uncertainty with a Fisher approximation.
• Ignoring parameter correlations (diagonalizing Cov[theta]); or skipping covariance shrinkage and encountering numerical instability.
• After optimizing in transform space, propagating to the physical domain without the Jacobian; or skipping non-dimensional scaling, causing scale mismatch in optimization and propagation.

XIII. Output Anchors and Citations

• Minimal equations: S71-1 (Delta linearization), S71-2 (arrival-time linearization), S71-3 (sampling-based propagation), S71-4 (predictive intervals), S71-5 (variance decomposition and contributions).
• Standard workflow: Mx-3 (uncertainty propagation pipeline).
• Interfaces: I30 7/8/10 and compare_param_sets.
• Cross-volume citations: EFT.WP.Core.Equations S20-* (paths & arrival time), S50-* (continuity & transport), S80-1 (statistical windows).