Marco Econométrico para el Análisis de Cointegración y Corrección de Errores | Determinación de Orden de Integración I(1) | Fundamento Teórico | Protocolo de Verificación | Justificación del Nivel de Significancia | Análisis de Cointegración | Enfoque Dual: Engle-Granger y Johansen | Procedimiento Engle-Granger con Phillips-Ouliaris | Test de Johansen con Traza | Justificación de la Regla "Either" | Modelo de Corrección de Errores (ECM) | Especificación del ECM Lineal | Selección Óptima de Rezagos | Test Unidireccional con Errores HAC | Hipótesis y Justificación | Inferencia Robusta | Extensión No-Lineal con MARS | Motivación Económica | Especificación del Modelo MARS-ECM | Configuración de Hiperparámetros | Validación Cruzada Temporal | Rolling-Origin con Ventana Deslizante | Parámetros de Configuración | Justificación de Sliding Window | Validación Cruzada Anidada | Métricas de Evaluación | Métricas en Escala | Métricas Relativas | Métrica Explicativa | Descomposición de Theil | Criterios de Estabilidad Temporal | Métrica de Soporte | Umbrales de Validación | Métricas Ajustadas por Estabilidad | Implementación Computacional | Paralelización Multi-Nivel | Gestión de Progreso | Complejidad Computacional

Econometric Framework for Cointegration and Error Correction Analysis | Determination of Integration Order I(1) | Theoretical Foundation | Verification Protocol | Justification of Significance Level | Cointegration Analysis | Dual Approach: Engle-Granger and Johansen | Engle-Granger Procedure with Phillips-Ouliaris | Johansen Trace Test | Justification of the "Either" Rule | Error Correction Model (ECM) | Linear ECM Specification | Optimal Lag Selection | Unidirectional Test with HAC Errors | Hypothesis and Justification | Robust Inference | Non-Linear Extension with MARS | Economic Motivation | MARS-ECM Model Specification | Hyperparameter Configuration | Temporal Cross-Validation | Rolling-Origin with Sliding Window | Configuration Parameters | Sliding Window Justification | Nested Cross-Validation | Evaluation Metrics | Scale-Dependent Metrics | Relative Metrics | Explanatory Metric | Theil Decomposition | Temporal Stability Criteria | Support Metric | Validation Thresholds | Stability-Adjusted Metrics | Computational Implementation | Multi-Level Parallelization | Progress Management | Computational Complexity

Marco Teórico de Modelos de Estado-Espacio para Series Temporales | Representación General Estado-Espacio | Formulación Matemática | Ventajas del Framework Estado-Espacio | Componentes Estructurales Implementados | Nivel Local (Local Level - LL) | Tendencia Lineal Local (Local Linear Trend - LLT) | Componente Estacional (Opcional) | Selección Bayesiana de Variables con Spike-and-Slab | Motivación Económica | Prior Spike-and-Slab | Calibración de Hiperparámetros | Inferencia Bayesiana mediante MCMC | Algoritmo de Muestreo | Configuración MCMC | Diagnósticos de Convergencia | Validación Temporal con 0[^5] | Esquema de Validación | Procedimiento por Fold | Métricas de Evaluación Predictiva | Expected Log Predictive Density (ELPD) | Métricas de Error Puntual | Métricas de Calibración | Selección de Estructura Óptima | Grid de Estructuras | Criterio de Selección | Criterio de Victoria y Estabilidad | Victoria por Fold | Métricas de Estabilidad | Implementación Computacional y Optimizaciones | Arquitectura de Procesamiento | Gestión de Casos Especiales | Estructura de Datos de Salida | Análisis Comparativo con Metodologías Previas | Tabla Comparativa de Enfoques | Ventajas Distintivas de BSTS | Limitaciones Relativas | Extensiones y Desarrollos Futuros | Mejoras Inmediatas | Extensiones Metodológicas | Pseudocódigo del Pipeline Completo | Conclusiones Metodológicas

State-Space Model Framework for Time Series | General State-Space Representation | Mathematical Formulation | Advantages of the State-Space Framework | Implemented Structural Components | Local Level (LL) | Local Linear Trend (LLT) | Seasonal Component (Optional) | Bayesian Variable Selection with Spike-and-Slab | Economic Motivation | Spike-and-Slab Prior | Hyperparameter Calibration | Bayesian Inference via MCMC | Sampling Algorithm | MCMC Configuration | Convergence Diagnostics | Temporal Validation with 0[^5] | Validation Scheme | Procedure per Fold | Predictive Evaluation Metrics | Expected Log Predictive Density (ELPD) | Point Error Metrics | Calibration Metrics | Optimal Structure Selection | Structure Grid | Selection Criterion | Victory and Stability Criterion | Victory per Fold | Stability Metrics | Computational Implementation and Optimizations | Processing Architecture | Special Case Management | Output Data Structure | Comparative Analysis with Previous Methodologies | Comparative Approach Table | Distinctive Advantages of BSTS | Relative Limitations | Extensions and Future Developments | Immediate Improvements | Methodological Extensions | Complete Pipeline Pseudocode | Methodological Conclusions

1. Purpose and Relation to the v08 Family* | 2. Setting and Notation | 2.1. AMM with $K \geq 1$ Distributional Slots | 2.2. Slot Conventions | 3. Sub-phase 8.3.1 -- param_specs as the Per-Slot Resolution Layer | 4. Sub-phase 8.3.2 -- The (D-ID) Per-Slot Identifiability Check (4C) | 5. Sub-phase 8.3.4 -- Custom Families and lognormal_loc_scale | 6. Sub-phase 8.3.5a -- Student-$t$ ($K = 3$, stan_id = 8) | 7. Sub-phase 8.3.5b -- Tweedie ($K = 3$, stan_id = 9) | 7.1. Hybrid lpdf: Dunn-Smyth series + saddlepoint | 7.2. THETA_REF_PRIOR_BLOCK per-family expansion | 7.3. Initial-value helper | 8. Sub-phase 8.3.6 -- Mixtures and Hurdle Models | 8.1. Algebraic decompositions | 8.2. Structural decisions | 8.3. Deferred cases | 9. Sub-phase 8.3.7 -- Heterogeneous Families per Slot | 10. Sub-phase 8.3.8 -- B-spline W Basis (D-D3) | 10.1. Cox-de Boor recurrence (D2 detailed) | 10.2. Boundary knot semantics | 11. Sub-phase 8.3.9 -- Residuals G1 / G2 / G3 + S3 Methods Re-validation + Route B predict.gdpar_fit for $K > 1$ | 11.1. (G1) Pearson-type residual per slot | 11.2. (G2) Randomized quantile residual (Dunn-Smyth) per slot | 11.3. (G3) DHARMa-compatible scaled residual per slot | 11.4. S3 method re-validation under $K > 1$ | 11.5. Bootstrapped goldens and the 16-column manifest | 12. Sub-phase 8.3.10 -- Release-Gate for Session 8.3 | 12.1. Fuzz testing protocol | 12.2. Debts recorded at release-gate | 13. Synthesis: The Per-Slot Architecture as a Coherent Design | 14. Implementation Status (Release-Gate Context) | 15. References (selective)

1. What this vignette covers | 2. Building a multivariate spec | 2.1. Direct construction with dimwise() | 2.2. Per-dimension overrides with override() | 2.3. Chainable construction with amm_build() | 2.4. Canonical serialisation: amm_save_spec() and amm_load_spec() | 3. Multivariate families: gdpar_family_multi() | 3.1. Construction | 3.2. Auto-promotion | 3.3. Constraints | 4. The modulating component for p > 1 | 4.1. Materialisation at a given p | 4.2. Splitting a separable basis with as_per_k() | 4.3. Why W stays top-level | 5. Calling gdpar() for p > 1 | 5.1. Outcome layout | 5.2. Parametrization aggregation | 5.3. Full call signature for the multivariate path | 6. Inspecting the pre-flight report | 7. Extracting predictions and coefficients | 7.1. predict() returns a 3-D array | 7.2. coef() returns a unified gdpar_coef object | 7.3. as.data.frame() for tidy pipelines | 8. End-to-end worked example | 9. Reporting parametrization decisions across scenarios | 9.1. Reading the CSV directly | 9.2. Hit-rate aggregation | 9.3. Faceted plot | 10. PSIS-LOO via gdpar_loo() | 10.1. Default aggregation: per-subject | 10.2. Diagnostic aggregation: per-cell | 10.3. Pareto-$k$ caveats | 10.4. Experimental status | 11. Known limitations | 11.1. Separable W only | 11.2. Per-coordinate heterogeneous families are a future block (per-slot heterogeneity is implemented as of 8.3.7) | 11.3. Single global sigma_W shared across blocks | 11.4. Multi-parametric extension is a future block | 11.5. Pre-flight wall-time | 12. References and cross-references

1. Purpose | 2. Setting and Notation | 2.1. The Hierarchical Bayesian Path 1 Model | 2.1.1. Two design regimes: random and conditional | 2.1.2. Generative assumption on the data | 2.2. Distance on the Parameter Space | 2.3. Notation Summary | 3. Three Asymptotic Layers | 4. Standing Asymptotic Hypotheses | 5. Theorem 4A: Posterior Consistency | 6. Theorem 4B: Posterior Contraction Rate | 6.1. Specialization to AMM Levels | 6.2. Numerical Verification of Contraction | 7. Theorem 4C: Bernstein-von Mises | 7.1. Semiparametric Bernstein-von Mises (Partial Result) | 8. Specialization to AMM Special Cases (Block 3) | 9. Open Questions | 10. Implementation Implications for Path 1 (Stan / cmdstanr) | 10.1. Prior Specification and (PRIOR-KL), (PRIOR-THICK) | 10.2. Convergence Diagnostics and (TEST), (SIEVE) | 10.3. Bernstein-von Mises Calibration Check | 10.4. (REG-EST) of Block 2 Specialized to Path 1 | 11. Summary | 12. Connections to Subsequent Blocks | Appendix A. Asymptotic Notation | Appendix B. Asymptotic Hypothesis Table | References Cited in This Block

1. Purpose | 2. Setting and Notation | 2.1. The Path 2 Model | 2.1.1. Two design regimes (parallel to Block 4) | 2.2. Vector vs. Scalar Convention for $\beta(\cdot)$ | 2.3. Distance on the Parameter Space | 2.3. Notation Summary | 3. Three Asymptotic Layers | 4. Standing Asymptotic Hypotheses for Path 2 | 5. Theorem 5A: Pointwise Consistency | 6. Theorem 5B: Uniform Consistency and Rate | 7. Theorem 5C: Pointwise Asymptotic Normality | 7.1. Scope of (L3): Explicit and Tight | 8. Proposition 5D: Adaptive Smoothing under Data-Driven $\widehat{\lambda}$ | 9. Specialization to Block 3 Special Cases | 10. Open Questions | 11. Implementation Implications for Path 2 (mgcv) | 11.1. Smoothing Parameter Selection | 11.2. Confidence Interval Reporting | 11.3. (REG-EST) of Block 2 Specialized to Path 2 | 12. Summary | 13. Connections to Subsequent Blocks | Appendix A. Asymptotic Notation for Path 2 | Appendix B. Asymptotic Hypothesis Table for Path 2 | References Cited in This Block

1. Purpose | 2. Setting and Notation | 2.1. The Path 3 Model | 2.1.1. Two design regimes (parallel to Blocks 4-5) | 2.2. Regimes of Asymptotic Analysis | 2.3. Distance and Notation | 3. Three Asymptotic Layers and Their Status for Path 3 | 4. Standing Asymptotic Hypotheses for Path 3 | 5. Theorem 6A: Consistency under the NTK Regime (Partial) | 6. Theorem 6B: PAC-Bayes Generalization Bound (Non-Asymptotic) | 7. Proposition 6C: Infinite-Width Gaussian Process Limit (Characterization of the Prior) | 8. Proposition 6D: Function-Level Consistency under Fixed Architecture (Partial) | 9. Specialization to Block 3 Special Cases | 10. Open Questions | 11. Implementation Implications for Path 3 (torch) | 11.1. Prior Specification (Bayesian Variant) | 11.2. PAC-Bayes Bound Reporting (Theorem 6B) | 11.3. NTK Diagnostic | 11.4. Discrimination Protocol Cross-Reference | 11.5. (REG-EST) of Block 2 Specialized to Path 3 | 12. Summary | 13. Connections to Subsequent Blocks | Appendix A. Asymptotic Notation for Path 3 | Appendix B. Asymptotic Hypothesis Table for Path 3 | References Cited in This Block

1. Purpose | 2. The Cognitive Starting Point: Driver Prediction Formalized | 3. Hierarchical Bayesian Cognition (Tenenbaum, Kemp, Griffiths, Goodman 2011) | 4. Core Knowledge as Structured Prior (Spelke and Kinzler 2007) | 5. The "Machines That Learn and Think Like People" Program (Lake, Ullman, Tenenbaum, Gershman 2017) | 6. Probabilistic Program Induction and One-Shot Generalization (Lake, Salakhutdinov, Tenenbaum 2015) | 7. Resource-Rationality (Lieder and Griffiths 2020) | 8. Mapping Cognitive-Science Strands to AMM Features | 9. What the Cognitive Analogy Does and Does Not Justify | 10. Open Questions on Cognitive Grounding | 11. Connections to the Rest of the Addendum | 12. Summary | References Cited in This Block

1. What this vignette covers | 2. Setup | 3. The grf adapter in three lines | 4. The EconML adapter | 4.1. One-time installation | 4.2. Running the EconML adapter | 4.3. Caveat: serialization of the EconML state | 5. Reading the output | 6. Writing your own adapter (DoubleML as an example) | 7. Troubleshooting (Python-side) | 8. Where to go next

1. What this vignette covers | 2. The temporal pair (recap) | 3. The spatial pair | 3.1. Diagnostic: Moran's I | 3.2. Robust SE: spatial block bootstrap | 3.3. The default block size, and why n^(1/4) | 3.4. Data-driven block size (block_size = "auto") | 4. The full-Bayes path | 5. Reading the result, and the caveats | References

1. What this vignette covers | 2. The K > 1 API | 2.1. Three equivalent input forms | 2.2. Choosing K | 2.3. End-to-end example: Gaussian K = 2 | 2.4. Custom K > 1 families via gdpar_family_custom_K() | 2.5. Prediction | 3. Residual diagnostics: G1 / G2 / G3 | 3.1. API | 3.2. Posterior predictive draws and PPC | 4. DHARMa integration (optional) | 4.1. API | 4.2. When to use DHARMa vs the built-in G2 | 5. Worked example: zero-inflated negative binomial (K = 3) | 6. Custom family registry: gdpar_family_custom() (K = 1) | 7. Known limitations and future work | References

1. Purpose and Relation to v07 | 1.1. Epistemic Posture and the Critical Audit of v07 §5 | 1.2. Reading Order | 2. Notation (Multivariate Extension) | 2.1. Multivariate Population Reference and Heterogeneous Slots | 2.2. Marginal Likelihood in $\mathbb{R}^p$ and per-Slot Tensor under $K > 1$ | 2.3. Sensitivity Jacobian of the Conditional Posterior | 2.4. Total Variation, Weak Metric, and Posterior Distance | 2.5. Notation Summary | 3. Standing Hypotheses (Multivariate Extension) | 3.1. (EB-MARG-ID)$_p$ -- Multivariate Marginal Identifiability | 3.2. (PRIOR-FB-WEAK)$_p$ -- Weak FB Prior in $\mathbb{R}^p$ | 3.3. (HIER-COMPLEX)$_p$ -- Multivariate Hierarchical Regularity | 3.4. Discussion: Why the Asterisked Hypotheses are Strictly Stronger | 4. Theorem 7A* -- First-Order Asymptotic Equivalence (Multivariate) | 4.1. Proof Sketch (Extension of v07's Reference Argument) | 4.2. Why Total Variation is Restricted to Regime A (Multivariate Refinement) | 4.3. Practical Implication of Theorem 7A* | 4.4. Reduction to v07's Theorem 7A under $(p, K) = (1, 1)$ | 4.5. Audit Trail: v07 §5 Reviewed Line by Line | 5. Proposition 7B* -- Higher-Order Coverage Discrepancy (Matrix Form) | 5.1. Derivation Sketch | 5.2. Reduction to v07's Scalar Proposition 7B under $p = \dim(\xi) = 1$ | 5.3. Effective Sensitivity Diagonalization (Practical Form) | 5.4. Operational Note and the eb_correction Argument | 6. Theorem 7C* -- Compound Decision Bound (Multi-Slot $K > 1$) | 6.1. Reduction to v07's Theorem 7C under $K = 1$ | 6.2. Two Regimes of Practical Use (Multi-Slot Refinement) | 6.3. Caveat (Squared-Error Loss; Coverage Picture Unchanged) | 6.4. Reference | 7. Proposition 7D* -- Substantial Discrepancy Conditions (Multivariate) | 7.1. Reduction to v07's Proposition 7D under $(p, K) = (1, 1)$ | 8. Recommendation by Scenario (Multivariate Extension) | 9. Open Questions | 9.1. (O1*-EBFB) Adaptive Choice between EB and FB Based on Data (Multivariate) | 9.2. (O2*-EBFB) Higher-Order Coverage Correction for Multivariate EB | 9.3. (O3*-EBFB) Behavior under Model Misspecification (Multivariate) | 9.4. (O4*-EBFB) Computational Competitiveness of FB at Moderate $n$ under $K \cdot p$ Large | 9.5. (O5*-EBFB) Numerical Anti-Fragility of Multivariate Laplace under Non-Conjugacy (New) | 10. Connections to Subsequent Blocks | 11. Joint Kernel Stein Discrepancy: the gdpar_ksd_joint() Helper | 11.1. The Stein Operator and the Stein Kernel | 11.2. Empirical Gaussian Target (B9.4 iteration) and Future Extension | 11.3. Base Kernel and Bandwidth | 11.4. ESS-Weighted Variant | 11.5. Operational Reading: gdpar_compare_eb_fb and gdpar_ksd_joint are Complementary | 12. Summary | Appendix A. Notation Correspondence: Multivariate vs.\ Scalar | Appendix B. Hypothesis Table (Multivariate) | References Cited in This Block

1. Purpose | 2. Setting and Notation | 2.1. Formal Specification of EB and FB | 2.2. Distance Between EB and FB Posteriors | 2.3. Notation Summary | 3. Three Layers of EB-vs-FB Comparison | 4. Standing Hypotheses for EB-vs-FB Comparison | 5. Theorem 7A: First-Order Asymptotic Equivalence of EB and FB | 6. Proposition 7B: Higher-Order Coverage Discrepancy of EB | 7. Theorem 7C: Finite-Sample Compound Decision Bound | 8. Proposition 7D: Conditions under which EB and FB Differ Substantially | 9. When to Use EB vs FB: Recommendation by Scenario | 10. Open Questions | 11. Implementation Implications for Path 1 | 11.1. Path 1: Stan Configuration | 11.2. Path 2: Penalty Parameter Selection (EB-like on a Different Object) | 11.3. Path 3: Variational Inference vs. Posterior Sampling | 11.4. (REG-EST) of Block 2 under EB and FB | 12. Summary | 13. Connections to Subsequent Blocks | Appendix A. EB-vs-FB Notation | Appendix B. EB-vs-FB Hypothesis Table | References Cited in This Block

Why this block exists | The conceptual bridge (geometry of the user's document ↔ sampling) | An organic critical reading of the source documents | The taxonomy: pathologies as a hierarchy of geometries | The diagnostic (RG.1): size-invariant signals | The engine and the ladder (RG.2--RG.4) | Level 0 -- Euclidean diagonal (the default) | Level 1 -- Euclidean dense | Level 3 -- Riemannian (Fisher / SoftAbs, and the learned GP-Fisher) | Level 4 -- Finsler / relativistic (heavy tails) | Level 5 -- sub-Riemannian (the count; "the glory") | The orchestrator (RG.5): diagnose → select → sample → re-diagnose → escalate or certify | Option A: the flat direction (RG.6 part i, D96) | Integration: the bridge and the one-call fit (RG.6 part ii) | RG.7: applying the capability to the real count, and the certified limit | The automated fallback (gdpar_geom_laplace(), laplace_fallback) | Honesty: demonstrated vs. conjectured, and no overreach | Reproducing the heavy runs

1. Purpose | 2. Gnoseological vs. Ontological Status of the Reference | 3. Three Structural Conditions | 3.1. (HOM) No Latent Stratification at the Parameter Level | 3.2. (REG) Regime Stability | 3.3. (CLOS) Operational Closure of the Regime | 4. Three Layers of Gnoseological Validity | 5. Proposition 2A: Algebraic-Functional Status of the Reference | 6. Lemma 2B: Statistical Consistency of the Empirical Reference Estimator | 7. Proposition 2C: Diagnostic Battery for the Three Conditions | 7.1. Test for (HOM): Latent Mixture in the Posterior of $\theta_i$ | 7.2. Test for (REG): Change-Point Detection on Fitted References | 7.3. Test for (CLOS): Extreme-Deviation Profile Inspection | 7.4. Joint Verdict | 8. When Conditions Fail: Operational Consequences | 9. Implementation Implications | 9.1. Path 1 (Hierarchical Bayesian via Stan) | 9.2. Path 2 (Varying-Coefficient via mgcv) | 9.3. Path 3 (Hypernetwork via torch) | 10. Summary | 11. Connections to Subsequent Blocks | Appendix A. Notation | Appendix B. Acronyms and Hypotheses | References Cited in This Block

1. What this vignette covers | 2. B-spline W bases | 2.1. When to switch from polynomial to B-spline | 2.2. API | 2.3. End-to-end example | 2.4. Diagnostics and known limitations | 3. Heterogeneous families per slot (K > 1) | 3.1. The default homogeneous case | 3.2. When to declare heterogeneous families | 3.3. API | 3.4. End-to-end example | 3.5. Identifiability and information diagnostics | 4. Combining B-spline W with heterogeneous families | 5. Known limitations and future work | References

1. What this vignette covers | 2. API | 2.1. Global flag | 2.2. Per-component overrides | 2.3. When to use which mode | 2.4. Anchor configuration | 3. The "auto" decision: three filters | 3.1. Filter 4 — Divergence attribution | 3.2. Filter 5 — E-BFMI | 3.3. Filter 6 — Info ratio (Path B') | 3.3.1. Effective coefficients and reference scale | 3.3.2. Log info ratio and block bootstrap | 3.3.3. Z-tests and thresholds | 4. Introspecting the decision | 5. Worked example | 6. Known limitations | 6.1. Confounding between a and W when covariates overlap | 6.2. Identifiability of individual basis components of W | 6.3. Block bootstrap versus naive bootstrap | 6.4. Pre-flight wall-time cost | 7. Summary recommendations | References

1. What this vignette covers | 2. API | 2.1. The group argument | 2.2. What changes inside the model | 3. Pre-flight condition (C7) | 4. Coefficient summary under grouping | 5. Cross-group prediction | 6. Interaction with the parametrization toggle | 7. Reproducibility checklist | 8. See also

1. Purpose | 2. Setting and Notation | 2.1. AMM and CATE/ITE: Formal Specifications | 2.2. The Object-Level Distinction | 2.3. Notation Summary | 3. Three Layers of Positioning | 4. Proposition 8A: The Conceptual Map | 5. Theorem 8B: The AMM-to-CATE Bridge as an Explicit Reparametrization | 6. Proposition 8C: Conceptual Placement of CATE Meta-Learners within AMM | 7. Proposition 8D: Decision-Theoretic Position and the "Fundamental Problem" | 8. Recommendation by Scenario | 9. Open Questions | 10. Implementation Implications | 10.1. Bridge invocation | 10.2. Reporting | 10.3. Comparison with meta-learner libraries | 11. Summary | 12. Connections to Subsequent Blocks | Appendix A. CATE/ITE Notation | Appendix B. CATE/ITE Hypothesis Table | References Cited in This Block

1. Motivation: From Cognition to Statistics | 1.1. An Empirical Observation | 1.2. The Cognitive Structure of the Process | 1.3. What Distinguishes This Process from a Standard Model | 2. General Mathematical Formulation | 2.1. Fundamental Equation | 2.2. Individual Prediction | 2.3. Components of the Framework | 3. Path 1: Hierarchical Bayesian Model with Covariate-Dependent Random Effects | 3.1. General Description | 3.2. Hierarchical Specification | 3.2.1. Population Level | 3.2.2. Individual Level | 3.2.3. Observation Level | 3.3. Extension for Zero Inflation | 3.3.1. Structural vs. Sampling Zeros | 3.3.2. Mixture Model with Individual Deviation | 3.4. Extension for Dependence Among Observations | 3.5. Multivariate Extension | 3.6. Recommended Implementation | 4. Path 2: Varying-Coefficient Models | 4.1. General Description | 4.2. Formulation | 4.3. Relation to the Population Reference | 4.4. Estimation of $\beta(\cdot)$ | 4.5. Strengths | 4.6. Limitations | 4.7. Extension for Zero Inflation | 4.8. Extension for Dependence | 5. Path 3: Conditional Parameter Networks (Hypernetworks / Amortized Inference) | 5.1. General Description | 5.2. Formulation | 5.3. Mechanism of Anchoring to the Population Reference | 5.4. Learning Process | 5.5. Extension for Zero Inflation | 5.6. Extension for Dependence | 5.7. Multivariate Extension | 5.8. Strengths | 5.9. Limitations | 5.10. Recommended Implementation | 6. Comparative Table of the Three Paths | 7. Selection Guide by Scenario | 8. Formal Summary | 8.1. Central Principle | 8.2. Canonical Equation | 8.3. Desirable Properties | 8.4. Universality across Data Conditions | Appendix: Notation

1. Who this vignette is for | 2. Installation | 3. A synthetic dataset | 4. A first fit | 5. Reading the print | 6. Coefficients | 7. Predictions | 8. Diagnostics | 9. What to read next | 10. Summary

1. What this vignette covers | 2. The four-layer snapshot schema | 2.1. Layer A — Structural (class signatures, shapes) | 2.2. Layer B — Discrete (bit-exact integer diagnostics) | 2.3. Layer C — Continuous (Monte Carlo standard error tolerance) | 2.4. Layer D — Sanity (absolute floors) | 2.5. The parametrization_resolved audit field | 3. End-to-end workflow | 3.1. Step 1 — Snapshot a reference fit | 3.2. Step 2 — Persist the snapshot | 3.3. Step 3 — Compare a fresh fit against the golden | 4. Tuning k_sigma and sanity_floor | 4.1. k_sigma: sensitivity of layer C | 4.2. sanity_floor: domain-specific absolute thresholds | 4.3. Combining k_sigma and sanity_floor | 5. Integration with testthat | 6. Workflow recommendations | 7. Known limitations | 7.1. Experimental status | 7.2. Stan implementation drift | 7.3. RNG seed semantics across hardware | 7.4. Layer C operates on posterior means only | 7.5. The structural layer is not exhaustive | 8. References and cross-references

1. Purpose | 2. Setting and Notation | 2.1. Reminder of (D-ID) and its concrete verification | 3. Theorem 3.1: Standard Linear Regression | 4. Theorem 3.2: Hierarchical Linear Model with Covariates in Random Effects | 4.1. Per-group anchor as random intercept (Block 6.5) | 5. Theorem 3.3: Random-Coefficient Model | 6. Theorem 3.4: Hastie-Tibshirani Varying-Coefficient Model | 7. Theorem 3.5: Reference-Modulated Varying-Coefficient Model | 8. Theorem 3.6: Hierarchical Bayesian with Multiplicative Interaction | 9. Proposition 3.7: Hypernetwork Models | 10. Summary of Subsumption Map | 11. Connections to Subsequent Blocks | Appendix A. Verification Checklist Template | References Cited in This Block

1. Purpose | 2. Setting and Notation | 3. The AMM Hierarchy: Stratifying the Space of Deviation Forms | 3.1. Level 0 -- Degenerate | 3.2. Level 1 -- Linear Additive | 3.3. Level 2 -- Canonical AMM | 3.4. Level 2.5 -- Full-Matrix Multiplicative | 3.5. Level 3 -- Quadratic Extensions | 3.6. Level $K$ -- Polynomial Closure | 3.7. Level $\infty$ -- Universal AMM | 3.8. Approximation Scheme: From Polynomial AMM to Continuous Functions | 4. Why Level 2 Is the Canonical Default | 5. Standing Assumptions | 6. Identifiability of Canonical AMM | 6.1. Three Layers of Identifiability | 6.2. The Functional Independence Condition | 6.3. Theorem 1A: Algebraic-Functional Identifiability | 6.3.1. Proof of Theorem 1A | Step 1: Marginal expectation identifies $\theta_*$. | Step 2: Subtract the common reference and form differences. | Step 3: Abstract FIC at $\theta_*$ forces each difference to zero. | 6.3.2. Necessity of FIC | 6.4. Lemma 1B: Statistical Bridge | 6.5. Corollary 1: Centering of the Framework | 6.6. Proposition 1C: Numerical Verifiability via the Gram Matrix | 6.6.1. Condition C4-bis: Cross-Coordinate Identifiability for p > 1 | 6.6.1.1. The failure mode | 6.6.1.2. The condition | 6.6.1.3. Why the extended Gram matrix cannot detect C4-bis | 6.6.1.4. The implementation: rigor = "full" vs rigor = "fast" | 6.6.1.5. Pedagogical example: overlap that fails (rigor = "full") | 6.6.1.6. Pedagogical example: coord-wise disjoint that passes | 6.6.1.7. Worked design pattern: when overlap is unavoidable | 6.6.1.8. Summary | 6.6.2. Condition C7: Anti-aliasing under per-group anchors (Block 6.5) | 6.6.2.1. The failure mode | 6.6.2.2. The condition | 6.6.2.3. Implementation | 6.6.2.4. Pedagogical example: factor(group) in a fails (C7) | 6.6.2.5. Worked design pattern: when a column is constant per group | 6.6.2.6. Summary | 6.7. Theorem 1E: Identifiability of $W$ on the Prior Support | 6.8. Proposition 1F: Scope and Limits of Identifiability for the Hypernetwork (Path 3) | 6.8.1. Empirical Protocol for Discriminating between Richer Structure and Non-Identifiability | 6.8.2. Decision Rule | 7. Identifiability of Higher Levels of the Hierarchy | 8. Standard Models as Special Cases | 8.1. Standard Regression | 8.2. Random-Coefficient Model | 8.3. Varying-Coefficient Model (Hastie and Tibshirani 1993) | 8.4. Hierarchical Bayesian with Multiplicative Interaction | 8.5. Full Canonical AMM | 8.6. Hypernetwork | 9. Test for Component Selection | 9.1. Bayesian Selection via Stratified PSIS-LOO | 9.2. Frequentist Selection via Generalized LRT | 9.3. Hypernetwork Component Norms | 9.4. Reported Selection Output | 10. Implementation Implications | 10.1. Path 1: Hierarchical Bayesian via Stan | 10.2. Path 2: Varying-Coefficient via mgcv | 10.3. Path 3: Amortized Inference via torch | 11. Summary of the Block | 12. Connections to Subsequent Blocks | Appendix A. AMM Notation | Appendix B. Acronyms and Hypotheses | References Cited in This Block

1. What this vignette covers | 2. Setup | 3. Minimal gdpar_eb() call (K = 1 + p = 1) | 4. Path A (K = 1, p > 1): multivariate outcome | 5. Path B (K > 1, p = 1): multi-parametric family | 6. Path C (K > 1, p > 1): full K x p extension | 7. Numerical diagnostics: how to read them | 8. EB vs FB comparison via gdpar_compare_eb_fb() | 9. Troubleshooting | 9.1. gdpar_eb_numerical_error: kappa = ... | 9.2. gdpar_unsupported_feature_error on Path C | 9.3. Multi-start dispersion warning | 9.4. Path B conditional HMC instability under logit-strict links | 9.5. Path C smoke validation | 10. Where to go next | References

1. Purpose and position in the package | 2. Notation inherited from v08 (rapid reference) | 3. Definition: the T-learner AMM-side | 4. Identification assumptions | 4.1. Inherited from v08 | 4.2. Residual no-confounding (T-learner-specific) | 4.3. What the bridge does not assume | 5. Estimation | 6. Inference: per-observation credible bounds and the ATE | 7. Identifiability per arm | 8. Minimum reproducible example | 9. Limitations of the T-learner | 10. Open questions (O-CATE)* | Appendix A. Notational correspondence with Kuenzel et al. (2019) | Appendix B. Implementation notes for future external comparators (8.5.B preview) | References cited in this addendum

1. Purpose and position in the package | 2. Notation inherited from v08 and v08b (rapid reference) | 3. The pluggable adapter contract | 4. Identification under cross-method comparison | 5. Concordance criterion | 6. Identifiability per arm under the bridge | 7. Limits of the comparison | 8. Minimum reproducible example (CRAN-valid, R-only) | 9. Open questions (O*-CMP) | Appendix A. Notational correspondence with the meta-learner literature | Appendix B. Implementation notes for additional adapters | References cited in this addendum

1. One disaggregation engine, not two | Where the engine is used here | 2. Leave-Cluster-Out | Why a cluster, not a single sector | The cluster map is pluggable | A documented fallback until the MIP arrives | Reading the statistical layer honestly

What this package does | Why "evidence-based": the F1–F6 history | The model (state-space, "Model A") | Identification, honestly (rigour by layers) | Two engines, one trade-off | A runnable example (closed-form engine) | The MCMC engine (sketch, not evaluated here) | Data note: comparing index vs index | Coupling to the nested OU

1. The economic question (§2.4 anchor) | 2. The two-level model | 3. Single source of truth and bit-exact equivalence (D-IMPL-1) | 4. Stability I: bounded reversion speeds (D-IMPL-2) | 5. Stability II: the enriched Level 2 (D-IMPL-9) | 6. Honest dispatch and the disaggregation coupling (D-IMPL-3, D-IMPL-4) | 7. Two-level diagnostics (D-IMPL-5, D-IMPL-6) | 8. Reconstruction with uncertain K (D-IMPL-7) | 9. Per-level richness switches and the comparative experiment (D-IMPL-8) | 10. Exposing the Level-2 richness: nu_p and sigma_p(t) (D-IMPL-10) | 11. Level 3: prices of production gravitate around values (D-IMPL-9.4, D-IMPL-10.1, D-IMPL-11) | 12. The wedge: why the value coupling is not spurious correlation (D-S12.3) | 13. The reversion speed kappa: interpretive framework (D-S12.4) | 14. Validation | 15. Caveats (anti-overreach) | 16. Block 9 validation: internal, external, and simulation-based calibration | 17. The logical verdict: why the "negative" external results corroborate the thesis | References

Introduction | Basic usage | Coupling significance (corrected null) | Convergence test | Methodological notes and design decisions | Reproducibility

Purpose | 1. Factorial simulation study | 1.1. Empirical size: the calibration story | 1.2. Coverage (sign-corrected) | 1.3. HMC diagnostics | 1.4. Pre-registered benchmark verdicts | 1.5. Reproducing the simulation | 2. Empirical benchmarks | 2.1. The persistence ladder | 2.2. Practical guidance | 3. Reproducing the empirical benchmarks | References

Overview | Quick start: long-run variance and fixed-b HAR | Block bootstrap | Adaptive conformal | Hierarchical Bayesian fit | Envelope across likelihoods | Decision: ROPE, Bayes factor, stacking | Explicit equivalence constants | References

Introduction | BOCPD: Bayesian Online Changepoint Detection | Prior Specification | Normal-Gamma Prior | Normal Prior with Known Variance | Poisson-Gamma Prior | Hazard Prior | Geometric Hazard | Constant Hazard | Negative Binomial Hazard | Posterior Analysis | Probability of Change at Each Point | Posterior Visualization | Run Length Distribution | Shiryaev-Roberts Procedure | Uncertainty Quantification | Credible Intervals | Existence Probability | Online Mode | Multivariate BOCPD | Comparison: BOCPD vs Frequentist | Advantages and Limitations | Advantages | Limitations | Best Practices | References

Overview | Installation | Quick Start | Visualizing Results | Available Methods | Frequentist Methods | Bayesian Methods | Types of Changes | Prior Specification (Bayesian Methods) | Evaluation | Method Comparison | Online Detection | Uncertainty Quantification | Next Steps

Introduction | When to Use Offline Detection | PELT: Pruned Exact Linear Time | Penalty Selection | Minimum Segment Length | Binary Segmentation | Wild Binary Segmentation | Detecting Different Types of Changes | Mean Changes | Variance Changes | Mean and Variance Changes | Visualization | Segment Analysis | Uncertainty Quantification | Evaluation Against Ground Truth | Comparing Methods | Best Practices | Tips for Difficult Cases | Closely Spaced Changepoints | Small Change Magnitudes | Many Changepoints

Introduction | Installation | Quick Start | Methodological Framework | Epistemological Foundation | Method Selection Guidance | Spectral Decomposition | Wavelet Multi-Resolution Analysis | Empirical Mode Decomposition | Bayesian HP-GC Filter | Bayesian Variable Selection | Regularized Horseshoe Prior | Interpreting Shrinkage Factors | Dimensionality Reduction | PCA with Bootstrap | Dynamic Factor Models | Unit Root Testing | Test Battery | Interpretation Framework | The Master Function | Automated Interpretation | Best Practices | 1. Start with Exploration | 2. Choose Methods Thoughtfully | 3. Validate with Domain Knowledge | 4. Handle Diagnostics | Computational Considerations | Memory and Time | Parallelization | Memory Management | References

Abstract | Introduction and Epistemological Framework | Motivation | Conceptual Mapping | Spectral Decomposition Methods | Wavelet Multi-Resolution Analysis | Theoretical Foundation | MODWT Implementation | Filter Selection | Scale-Period Correspondence | Empirical Mode Decomposition | Sifting Algorithm | Reconstruction Property | Bayesian HP-GC Filter | State Space Formulation | Prior Configurations | Bayesian Sparse Regression: Regularized Horseshoe | Model Specification | Horseshoe Prior | Regularized Horseshoe | Shrinkage Factors | Effective Number of Non-Zeros | Global Shrinkage Calibration | Stan Implementation | Dimensionality Reduction | PCA with Bootstrap Significance | Standard PCA | Bootstrap Significance Testing | Entropy of Loadings | Dynamic Factor Models | Model | Factor Selection via Information Criteria | Unit Root Testing | Test Battery | ADF Test | ERS Tests | DF-GLS | Point Optimal Test | KPSS Test | Synthesis Algorithm | Master Function: signal_analysis() | Pipeline Architecture | Output Structure | Interpretation Framework | Signal Characteristics | Sparsity Assessment | Information Topology | Computational Considerations | Complexity Analysis | Parallelization | CRAN Compliance Notes | Soft Dependency: cmdstanr | Namespace Hygiene | References

Introduction | Installation | Quick Start Example | Step 1: Load Data and Compute Derivatives | Step 2: Explore the Data | Step 3: Fit the Equation | Step 4: Validate the Model | Step 5: Generate Output | Detailed Workflow | A. Numerical Differentiation | Total Variation Regularization (TVR) | Other Methods | Automatic Method Selection | B. Visual Exploration | C. Equation Discovery | Automated Symbolic Search | Theory-Guided Fitting | Julia Backend (Optional) | D. Residual Analysis and SDE Construction | Diagnostics | Modeling the Diffusion | Iterative GLS Estimation | E. Validation | Cross-Validation | Trajectory Simulation | Qualitative Behavior | Comprehensive Validation | F. Output and Reporting | LaTeX Equations | Tables | Full Report | Analysis Summary | Best Practices | Differentiation | Equation Discovery | Validation | Reporting | Further Reading | Session Info

Overview | Getting Started | Creating Example Data | Preparing Panel Data | Fitting Models | Two-Way Fixed Effects | Mundlak CRE Model | Model Comparison | Cross-Validation | Structural Break Tests | Full Analysis Pipeline

Marco Bayesiano para Modelado de Series Temporales con Estructura Autorregresiva | Especificación del Modelo Bayesiano con AR(1) | Estructura del Modelo Completo | Modelo Base de Referencia | Especificación de Priors | Validación Cruzada Temporal con Leave-Future-Out (LFO) | Implementación del Rolling-Origin con Ventana Deslizante | Criterios de Evaluación Predictiva | Expected Log Predictive Density (ELPD) | Root Mean Square Error (RMSE) y Métricas Clásicas | Criterio de Victoria por Fold | Inferencia Bayesiana con Hamiltonian Monte Carlo | Configuración del Muestreador NUTS | Diagnósticos de Convergencia | Determinación de Estabilidad Temporal y Soporte | Métrica de Soporte | Umbrales de Robustez | Interpretación del Soporte | Análisis Comparativo con Metodología ECM-MARS | Diferencias Fundamentales | Ventajas del Enfoque Bayesiano | Limitaciones Relativas | Implementación Técnica y Optimizaciones | Gestión de Memoria y Paralelización | Manejo de Casos Degenerados | Pseudocódigo del Pipeline Completo | Notas Técnicas y Consideraciones Especiales | Escalamiento y Estabilidad Numérica | Interpretación de Métricas Negativas | Extensiones Potenciales | Conclusiones Metodológicas

Bayesian Framework for Time Series Modeling with Autoregressive Structure | Specification of the Bayesian Model with AR(1) | Complete Model Structure | Reference Baseline Model | Prior Specification | Temporal Cross-Validation with Leave-Future-Out (LFO) | Rolling-Origin Implementation with Sliding Window | Predictive Evaluation Criteria | Expected Log Predictive Density (ELPD) | Root Mean Square Error (RMSE) and Classical Metrics | Victory Criterion per Fold | Bayesian Inference with Hamiltonian Monte Carlo | NUTS Sampler Configuration | Convergence Diagnostics | Determination of Temporal Stability and Support | Support Metric | Robustness Thresholds | Support Interpretation | Comparative Analysis with ECM-MARS Methodology | Fundamental Differences | Advantages of the Bayesian Approach | Relative Limitations | Technical Implementation and Optimizations | Memory Management and Parallelization | Handling Degenerate Cases | Complete Pipeline Pseudocode | Technical Notes and Special Considerations | Scaling and Numerical Stability | Interpretation of Negative Metrics | Potential Extensions | Methodological Conclusions