Overview | 1. Spatial Fay-Herriot Model (eblup_sfh) | Model Formulation | Fitting the Model | Unsampled Domains and Spatial Kriging | Multi-Threaded Bootstrap MSE | 2. Spatio-Temporal Fay-Herriot Model (eblup_stfh) | Fitting the Spatio-Temporal Model | Unsampled Domains and Missing Periods | 3. Comparing Models with autoplot | 4. Bayesian Spatio-Temporal Modeling (hb_area) | References

Overview | Model Formulation | Step-by-Step Example | 1. Data Preparation | 2. Fit BHF Model with Bootstrap MSE | 3. Inspect Domain Estimates | 4. Visualizing Results with autoplot | References

Introduction | Supported Distribution Families | Mathematical Formulation | Level 1: Sampling Model (Likelihood) | Level 2: Prior on Fixed Effects | Level 3: Prior on Random Effects | INLA Approximation | Spatial and Spatio-Temporal Structures | Spatial Priors | Temporal Models | Space-Time Interaction Structures (st_interaction) | Case Study: Spatio-Temporal Beta SAE on Emilia-Romagna Poverty | 1. Load Data and Construct Spatial Adjacency Matrix | 2. Fit Spatio-Temporal Beta Model | 3. Inspect Posterior Predictions and Uncertainty | Comparison and Validation Against tipsae (Stan MCMC) | Interactive Scalability Explorer ($n = 30$ to $n = 1,000$) | Inferential Equivalence & Validation Metrics ($n = 1,000$) | Cross-Sectional Count Data: Poisson & Negative Binomial | References

Introduction | Core Diagnostic Criteria | Practical Example: Diagnosing a Fay-Herriot Model | 1. Fit Model and Run diagnose() | 2. Inspect Diagnostic Components | Visual Diagnostics with autoplot() | 1. Calibration Plot (type = "calibration") | 2. Precision Comparison (type = "rse") | 3. Residual Diagnostics (type = "residuals" & type = "qq") | 4. Combined Diagnostic Dashboard (type = "all") | Diagnosing Bayesian Spatio-Temporal Models | References

Executive Summary | Benchmark Results Table | Interactive Benchmark Explorer | Architectural Insights: Why is fastsae so Fast? | Bayesian Spatio-Temporal Benchmark: fastsae (INLA) vs tipsae (Stan MCMC) | Interactive Beta SAE Benchmark Explorer ($n = 30$ to $n = 1,000$) | Inferential Equivalence & Validation Metrics ($n = 1,000$) | Finite Population Simulation & Ground Truth Validation

Introduction: Why Finite Population Simulation? | Simulation Architecture | Mathematical Formulations of Empirical Properties | Empirical Benchmark Results ($R = 100$ Replications, $D = 60$) | Key Empirical Findings: | National-Scale Benchmark ($D = 500$ Domains, $N \approx 250{,}000$ Individuals, $R = 25$) | Key National Scalability Takeaways: | Visualizations | 1. Domain-Specific Empirical RMSE | 2. Empirical 95% Credible Interval Coverage | How to Replicate | References

Introduction | The Fay-Herriot Model | Step-by-Step Example | 1. Load Package and Dataset | 2. Fit Fay-Herriot Model (eblup_fh) | 3. Model Summary and Coefficients | 4. Diagnostic Plots (autoplot) | Direct Estimates vs EBLUP | Mean Squared Error (MSE) Across Domains | Exact Numerical Equivalence with sae | References

Executive Summary | Benchmark Results Table | Interactive Benchmark Explorer | Architectural Insights: Why is fastsae so Fast?

Overview | 1. Spatial Fay-Herriot Model (eblup_sfh) | Model Formulation | Fitting the Model | Unsampled Domains and Spatial Kriging | Multi-Threaded Bootstrap MSE | 2. Spatio-Temporal Fay-Herriot Model (eblup_stfh) | Fitting the Spatio-Temporal Model | 3. Comparing Models with autoplot | References

Overview | Model Formulation | Step-by-Step Example | 1. Data Preparation | 2. Fit BHF Model with Bootstrap MSE | 3. Inspect Domain Estimates | 4. Visualizing Results with autoplot | References

Introduction | The Fay-Herriot Model | Step-by-Step Example | 1. Load Package and Dataset | 2. Fit Fay-Herriot Model (eblup_fh) | 3. Model Summary and Coefficients | 4. Diagnostic Plots (autoplot) | Direct Estimates vs EBLUP | Mean Squared Error (MSE) Across Domains | Exact Numerical Equivalence with sae | Next Steps | References

Introduction | Method Overview | Required Packages | Example: Income Estimation Using Linear Regression | Example: Binary Outcome Using Logistic Regression | Example: LightGBM with Hyperparameter Tuning | Supported Models | References | Conclusion