03-EFT.WP.Core.Parameters v1.0 | Chapter 6 — Calibration and Inference Workflow

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Chapter 6 — Calibration and Inference Workflow

I. Aims and Scope

• Unify the mathematical conventions and interface workflows for MLE, MAP, MCMC, and joint calibration, ensuring seamless integration with I30 5, I30 6, and I30 10.
• Provide evaluation rules and early-stopping strategies for information criteria and cross-validation, forming a reproducible experimental pipeline Mx-2.
• Typical coupling scenario: parameter calibration based on arrival time T_arr = ( ∫ ( n_eff / c_ref ) d ell ), with the path gamma(ell) and the measure d ell declared explicitly.

II. Inputs, Outputs, and Conventions

1. Input set
   • Model: y = f(x; theta) or a constrained form R(u, theta) = 0 with an observation map y = G(u).
   • Data: data = { (x_k, y_k) }_{k=1..N }, noise covariance Σ, weight matrix W = Σ^{-1}.
   • Priors and bounds: prior(theta), lb ≤ theta ≤ ub, C_eq(theta)=0, C_ineq(theta) ≤ 0.
2. Output set
   • Point estimates: theta_MLE, theta_MAP; posterior samples: { theta^{(s)} }_{s=1..S}.
   • Diagnostics: Fisher(theta_hat), Cov[theta], Corr[theta], information-criteria and cross-validation scores.
   • Report artifacts: non-dimensionalized parameter table, convergence curves, residuals, and sensitivity summaries.

III. Maximum Likelihood (MLE) Objective and Constraints (Minimal Equation S61-1)

1. Weighted negative log-likelihood
   • l(theta) def= - log L(data | theta) = ( 1 / 2 ) * Σ_{k=1..N} ( r_k(theta)^T * W_k * r_k(theta) ) + const
   • Residuals: r_k(theta) = y_k - f(x_k; theta)
2. Constrained MLE
   minimize_theta l(theta) subject to lb ≤ theta ≤ ub, C_eq(theta)=0, C_ineq(theta) ≤ 0
3. Gradients and Hessian (Gaussian approximation)
   • ∂l/∂theta = - J^T * W * r
   • ∂^2 l/∂theta^2 approx J^T * W * J（ignore second-order ∂J/∂theta）

IV. Maximum A Posteriori (MAP) and Regularization (Minimal Equation S61-2)

1. Posterior and negative log-posterior
   • post(theta | data) def= L(data | theta) * prior(theta) / Z
   • U(theta) def= - log post(theta | data) = l(theta) - log prior(theta) + const
2. Typical prior examples
   • prior(theta_i) = Normal(mu_i, sigma_i) ⇒ - log prior(theta_i) = ( 1 / 2 ) * ( (theta_i - mu_i)^2 / sigma_i^2 ) + const
   • Structured sparsity: prior(theta_i) = Laplace(b) ⇒ - log prior(theta_i) = |theta_i| / b + const
3. MAP optimization
   minimize_theta U(theta) subject to lb ≤ theta ≤ ub, C_eq=0, C_ineq ≤ 0

V. Bayesian Sampling (MCMC) and Posterior Estimation (Minimal Equation S61-3)

1. Target density: π(theta) ∝ exp( - U(theta) )
2. Sampling strategies
   • Gradient-driven: NUTS/HMC, requiring ∂U/∂theta (assembled from compute_jacobian in I30 10 plus prior gradients).
   • When gradients are hard to obtain: Random-Walk Metropolis or Adaptive Metropolis (less efficient).
3. Posterior statistics
   • E[theta_i | data] approx ( 1 / S ) * Σ_{s=1..S} theta_i^{(s)}
   • Cov[theta] approx covariance( { theta^{(s)} } )
   • Intervals: CI_{1-α}[theta_i] via sample quantiles or normal approximation.

VI. Multi-Dataset Joint Calibration (Minimal Equation S61-4)

1. Data clusters data = { data_m }_{m=1..M } with weights w_m ≥ 0:
   • log L_joint(data | theta) = Σ_{m=1..M} w_m * log L_m(data_m | theta)
   • U_joint(theta) = - log L_joint - log prior(theta) + const
2. Weighting strategies
   • Homoscedastic noise: w_m = 1
   • Heteroscedastic noise: w_m = 1 / σ_m^2 or sample-normalized w_m = n_m / Σ n_m
3. Scenario governance: record { data_m, w_m } and versions in create_scenario for reproducibility.

VII. Arrival-Time Coupling Conventions (Aligned with S20-)

• If observations are arrival times y_k = T_arr( gamma_k; theta ):
  T_arr( gamma_k; theta ) = ( ∫_{gamma_k} ( n_eff(x; theta) / c_ref(theta) ) d ell )
• Residuals
  r_k(theta) = y_k - ( ∫_{gamma_k} ( n_eff / c_ref ) d ell )
• Gradients (see S51-4)
  ∂r_k/∂theta_i = - ( ∫_{gamma_k} ( ( ∂ n_eff / ∂theta_i ) / c_ref - n_eff * ( ∂ c_ref / ∂theta_i ) / ( c_ref^2 ) ) d ell )
• Computation interfaces
  Path discretization discretize_path, quadrature via propagate_time, derivatives assembled by compute_jacobian.

VIII. Information Criteria and Cross-Validation (Minimal Equation S61-5)

1. With θ_hat a point estimate, p = dim(theta_free), N = sample size.
2. AIC and BIC
   • AIC = 2 * p - 2 * log L(data | θ_hat)
   • BIC = p * log(N) - 2 * log L(data | θ_hat)
3. WAIC (posterior mean log predictive density)
   • lppd = Σ_{k=1..N} log( ( 1 / S ) * Σ_{s=1..S} p(y_k | theta^{(s)}) )
   • p_waic = Σ_{k=1..N} Var_{s}( log p(y_k | theta^{(s)}) )
   • WAIC = -2 * ( lppd - p_waic )
4. LOO-CV (approximate)
   ELPD_LOO = Σ_{k=1..N} log( p_{-k}(y_k) )（Pareto-smoothed importance approximation可用）
5. Early-stopping criteria (validation set val)
   If Δ log L(val | θ_t) < ε_ll for K consecutive rounds, or Corr[theta]_t degrades persistently, trigger early stop.

IX. Non-Dimensionalization and Transform Strategy (Aligned with Chapter 3)

• Transform parameters to the real line before optimization
  phi_i = T_map(theta_i) such as log, logit, softplus
• Objective in phi space
  U_phi(phi) = U( T_map^{-1}(phi) ) - log | det( ∂ T_map^{-1} / ∂ phi ) |
• Baseline scaling for data and outputs
  bar_y = y / y0，bar_t = t / t0，bar_L = L / L0; proceed to optimization only after check_dim(expr) passes.

X. Calibration Pipeline Mx-2 (Standard Steps)

• Data curation: split train/val/test, construct Σ and W, declare windows avg_t/avg_V/avg_gamma.
• Parameter registration: register_param, set bounds/constraints/prior/transform, form theta_free.
• Identifiability pre-check: fisher_information and check_identifiability; if needed, tie_params or freeze.
• Initialization: sample_prior and heuristics (median or zscore standard point in the non-dimensional domain).
• MLE warm-start: minimize l(theta) to obtain theta_MLE for initializing MAP and MCMC.
• MAP refinement: minimize U(theta), record Cov approx F^{-1} and Corr.
• Posterior sampling: posterior_sample_mcmc (NUTS recommended); compute ELPD_LOO/WAIC; produce CIs.
• Joint calibration: if multiple datasets, minimize U_joint or use mixture likelihoods within MCMC.
• Early stop & selection: decide via validation log L, information criteria, and error thresholds; produce the best scenario snapshot.
• Reporting & regression: export_params("yaml") and compare_param_sets to establish the regression baseline; record Mx-2 metadata.

XI. Implementation Binding and Minimal Working Examples (I30 5 Family)

1. Point estimates
   • theta_mle = infer_mle(model, data_train, params=theta_free)
   • theta_map = infer_map(model, data_train, params=theta_free)
2. Posterior
   samples = posterior_sample_mcmc(model, data_train, params=theta_free, n=S, burn=B, method="NUTS")
3. Joint
   calibrate_joint(datasets=[D1,...,DM], weights=[w1,...,wM], params=theta_free)
4. Diagnostics
   fisher_information(model, theta_map, data_train)；regularize_cov(Cov, "shrinkage", alpha)
5. Cross-volume gradients
   J = compute_jacobian(eqn=S20_arrival, params=[...]) plugged into infer_* for first- and second-order information.

XII. Misuse and Conflict Checklist

• Never interchange n and n_eff; all arrival-time formulas must be ( ∫ ( n_eff / c_ref ) d ell ) with gamma(ell) declared.
• Do not omit the transform Jacobian: when optimizing MAP in phi-space, you must include log |det( ∂ T_map^{-1} / ∂ phi )|.
• An incorrect noise model in W induces biased estimates; if unknown, jointly calibrate σ or adopt robust likelihoods.
• Do not use normalized sensitivity near f(x; theta) ≈ 0; use a reference scale or truncation.
• In multi-dataset calibration, weights must not double-count samples; normalize explicitly or model heterogeneity in the likelihood.

XIII. Output Anchors and Citations

• Minimal equations: S61-1 (MLE negative log-likelihood), S61-2 (MAP negative log-posterior), S61-3 (posterior sampling density), S61-4 (joint calibration), S61-5 (information criteria and cross-validation).
• Standard workflow: Mx-2 (calibration pipeline).
• Interfaces: I30 5 (inference & calibration), I30 6 (identifiability), I30 10 (cross-volume Jacobian).
• Cross-volume citations: see EFT.WP.Core.Equations S20-* (arrival time), S40-* (tension field), and Chapter 3 of this volume (transforms).