Index: Computational Workflow

Computational Workflow

Routing Summary

Chapters 11–13 and 15 of Gelman, Vehtari & McElreath (2026): what HMC is actually doing, how to read its diagnostics, the catalogue of failure modes and their fixes, the approximate-inference ladder, and modeling as software development. 16 notes.

• Need to know what the sampler is exploring? → The Typical Set and the Log Posterior Density
• Need initialization / warmup / adaptation? → Initial Values, Adaptation, and Warmup
• Need to read $\hat{R}$ and ESS? → Chains, Iterations, and Effective Sample Size
• Need MCSE and how many digits to report? → Effective Sample Size and Monte Carlo Standard Error, How Many Digits to Report
• Need the fail-fast development loop? → Fit Fast, Fail Fast
• Need the catalogue of failure modes (funnels, aliasing, multimodality, divergences)? → Failure Modes and Steps Forward
• Need to fix computation by changing the model? → Modeling Ideas to Address Computing Problems
• Need the diagnostic ladder for non-convergence? → What to Do About Convergence Problems
• Need approximate inference? → Approximate Algorithms and Approximate Models, Variational Inference and Pathfinder, Approximations Based on Joint and Conditional Posterior Modes
• Need amortized / divide-and-conquer methods? → Simulation-Based and Amortized Inference, Divide-and-Conquer Algorithms
• Need engineering practice for models? → Statistical Modeling as Software Development

Concept Map

Concept	Note	Type	Depends On	Key Result
Typical set	The Typical Set and the Log Posterior Density	definition	—	High-dimensional mass is in a thin shell, not at the mode
Initialization and warmup	Initial Values, Adaptation, and Warmup	concept	The Typical Set and the Log Posterior Density	There is no universal default initial point
$\hat{R}$ and ESS	Chains, Iterations, and Effective Sample Size	definition	Initial Values, Adaptation, and Warmup	$\hat{R}<1.01$ comfortable; $\hat{R}>2$ means not mixing
MCSE	Effective Sample Size and Monte Carlo Standard Error	theorem	Chains, Iterations, and Effective Sample Size	Eq. 11.1–11.2; MCSE $=\text{sd}/\sqrt{\text{ESS}}$
Reporting precision	How Many Digits to Report	concept	Effective Sample Size and Monte Carlo Standard Error	Report only digits MCSE supports
Fail-fast loop	Fit Fast, Fail Fast	concept	—	Short runs and approximations to detect problems early
Failure modes	Failure Modes and Steps Forward	concept	Fit Fast, Fail Fast	Figures 12.4–12.12; funnels, aliasing, multimodality, the folk theorem
Model changes as computational fixes	Modeling Ideas to Address Computing Problems	concept	Failure Modes and Steps Forward	Figures 12.13–12.15; reparameterization, constraints, stronger priors
Convergence remedies	What to Do About Convergence Problems	concept	Failure Modes and Steps Forward	The ordered diagnostic ladder
Approximation ladder	Approximate Algorithms and Approximate Models	concept	Fit Fast, Fail Fast	Figure 13.1; approximating the algorithm vs. the model
Modal approximations	Approximations Based on Joint and Conditional Posterior Modes	concept	Approximate Algorithms and Approximate Models	Laplace; the joint mode’s pole at $\sigma =0$
VI and Pathfinder	Variational Inference and Pathfinder	definition	Approximate Algorithms and Approximate Models	Eq. 13.1–13.2; L-BFGS path, KL-best normal, importance resampling
Amortized inference	Simulation-Based and Amortized Inference	concept	Approximate Algorithms and Approximate Models	Train once, infer many times
Divide and conquer	Divide-and-Conquer Algorithms	concept	Approximate Algorithms and Approximate Models	Partition data, combine posteriors
Simpler models for computation	Fitting Simpler Models for Computational Purposes	concept	Approximate Algorithms and Approximate Models	Simplify to diagnose, then restore
Software practice	Statistical Modeling as Software Development	concept	—	Version control, modularity, testing, reproducibility

Notes

• The Typical Set and the Log Posterior Density — CONTAINS: the typical set; lp__ as a diagnostic; Figure 11.1; Ch. 11 intro, 11.1
• Initial Values, Adaptation, and Warmup — CONTAINS: Stan’s default init, when to override; Ch. 11.2–11.3
• Chains, Iterations, and Effective Sample Size — CONTAINS: $\hat{R}$, bulk/tail ESS, thresholds; Figure 11.2; Ch. 11.4
• Effective Sample Size and Monte Carlo Standard Error — CONTAINS: Eq. 11.1–11.2; Ch. 11.5
• How Many Digits to Report — CONTAINS: reporting rules; Figure 11.3; Ch. 11.6–11.8
• Fit Fast, Fail Fast — CONTAINS: the development loop; Figures 12.1–12.3; Ch. 12.1–12.2
• Failure Modes and Steps Forward — CONTAINS: the funnel, additive and multiplicative aliasing, label switching, the folk theorem; Figures 12.4–12.12; Ch. 12.3
• Modeling Ideas to Address Computing Problems — CONTAINS: reparameterization, sum-to-zero constraints, informative hyperpriors; Figures 12.13–12.15; Ch. 12.4
• What to Do About Convergence Problems — CONTAINS: the diagnostic order; Ch. 12.5–12.6
• Approximate Algorithms and Approximate Models — CONTAINS: Figure 13.1; the taxonomy; Ch. 13 intro, 13.6
• Approximations Based on Joint and Conditional Posterior Modes — CONTAINS: Laplace, marginal vs. joint mode; Ch. 13.1
• Variational Inference and Pathfinder — CONTAINS: ADVI, Pathfinder, multi-Pathfinder, Pareto $\hat{k}$ validation; Eq. 13.1–13.2; Ch. 13.2
• Simulation-Based and Amortized Inference — CONTAINS: neural posterior estimation; Ch. 13.3
• Divide-and-Conquer Algorithms — CONTAINS: data partitioning and posterior combination; Ch. 13.4
• Fitting Simpler Models for Computational Purposes — CONTAINS: simplify-to-diagnose; Ch. 13.5
• Statistical Modeling as Software Development — CONTAINS: version control, modularity, testing, reproducibility; Ch. 15

Sources

• Gelman Vehtari McElreath 2026 - Bayesian Workflow (book) — Chapters 11–13, 15, pp. 193–260

See Also

• Simulation-Based Calibration — validating that the computation is correct, not just converged
• Evaluating and Comparing — checking the model rather than the computation
• Sampling Problems with Latent Variables - No Vehicles in the Park — the aliasing and parameterization case study
• Challenge of Multimodality - Differential Equation for Planetary Motion — the multimodality and Pathfinder case study
• Debugging a Model - World Cup Football — a debugging session end to end