Part VII. Advanced topics

Part VII

This part covers the mature edges of Bayesian workflow: model uncertainty, robustification, design under uncertainty, communication, and capstone-level synthesis.

Part overview

Part VII extends the core workflow into the topics that distinguish mature Bayesian practice from elementary posterior calculation: model uncertainty, robustness, experimental design, communication, and open-ended projects.

Part learning goals

By the end of Part VII, learners should be able to:

  • compare models using predictive and marginal-likelihood criteria with appropriate caution;
  • formulate robust alternatives under misspecification and outliers;
  • reason about design under uncertainty and utility;
  • communicate Bayesian results responsibly to scientific collaborators.

Fully written chapters in this repository

Reading path

Recommended order:

  1. Model selection and averaging
  2. Robust Bayesian methods
  3. Bayesian experimental design
  4. Communication and interpretation
  5. Capstone projects

Chapter 27. Model selection and averaging

Core question

How should we compare or combine models when model uncertainty itself matters?

Section blueprint

  • Bayes factors and marginal likelihood.
  • Bridge sampling.
  • Bayesian model averaging.
  • Stacking.
  • Projection predictive selection.

Derivations that must appear

  • Marginal likelihood identity.
  • Posterior model probabilities under prior model weights.
  • Predictive-risk intuition for stacking.

Computational lab

  • Estimate a marginal likelihood with bridge sampling.
  • Compare PSIS-LOO model ranking with Bayes factors under prior sensitivity.

Common pitfalls

  • Using Bayes factors with vague or improper priors.
  • Confusing explanatory and predictive goals in model comparison.
  • Treating one selection method as universally optimal.

Exercises

  • Show how posterior model probabilities depend on prior model probabilities.
  • Compare BMA and stacking conceptually on a misspecified model set.

Core references

Full chapter

Chapter 28. Robust Bayesian methods

Core question

How can Bayesian models remain useful when data contain outliers or the likelihood is misspecified?

Section blueprint

  • Heavy-tailed priors.
  • Heavy-tailed likelihoods.
  • Contamination models.
  • Outlier indicators and latent-mixture robustness.
  • Misspecification and posterior concentration issues.

Derivations that must appear

  • Student-\(t\) likelihood as a scale-mixture intuition.
  • Contamination-model factorization.
  • Influence of heavy-tailed priors on posterior shrinkage.

Computational lab

  • Compare Gaussian and Student-\(t\) regression under outliers.
  • Fit a contamination mixture and inspect posterior outlier probabilities.

Common pitfalls

  • Overusing robustness as a substitute for understanding the data-generating process.
  • Ignoring the computational costs of mixture-based robustification.

Exercises

  • Show why a Student-\(t\) model downweights outliers more than a Gaussian model.
  • Compare posterior predictive behavior under robust and non-robust likelihoods.

Core references

Full chapter

Chapter 29. Bayesian experimental design

Core question

How should one choose data collection strategies when uncertainty and utility are both explicit?

Section blueprint

  • Utility-based design.
  • Expected information gain.
  • Sequential and adaptive design.
  • Active learning overview.
  • Computational challenges in design optimization.

Derivations that must appear

  • Expected utility criterion for design selection.
  • Mutual-information intuition for information gain.
  • Posterior update within a sequential design loop.

Computational lab

  • Compare fixed and adaptive sampling in a toy binary-outcome experiment.
  • Approximate design utility by Monte Carlo.

Common pitfalls

  • Confusing parameter uncertainty reduction with decision utility.
  • Ignoring design-stage prior sensitivity.

Exercises

  • Write a Monte Carlo estimator for expected utility of a design.
  • Compare two candidate designs under different utility functions.

Core references

Full chapter

Chapter 30. Communication and interpretation

Core question

How should Bayesian conclusions be reported so that uncertainty, model assumptions, and decision relevance are all clear?

Section blueprint

  • Reporting posterior summaries.
  • Interval interpretation.
  • Effect sizes and uncertainty communication.
  • Decision-oriented summaries.
  • Visual communication.
  • Reproducibility and literate analysis.

Derivations and conceptual targets

  • Translate posterior distributions into scientifically meaningful contrasts.
  • Explain posterior probability statements without frequency misuse.
  • Distinguish predictive uncertainty from parameter uncertainty in communication.

Computational lab

  • Build a one-page results summary from posterior draws.
  • Compare multiple interval types and visual summaries.

Common pitfalls

  • Reporting only posterior means.
  • Hiding assumptions behind polished plots.
  • Interpreting high posterior probability as practical importance.

Exercises

  • Rewrite a technical posterior summary for a nontechnical collaborator.
  • Build a plot that separates posterior uncertainty from observed-data noise.

Core references

  • Gabry et al. (2019)
  • Gelman et al. (2013)

Full chapter

Chapter 31. Capstone projects

Core question

How do all components of Bayesian workflow combine in a real, open-ended project?

Section blueprint

  • Project scoping and question formulation.
  • Data auditing and exploratory analysis.
  • Prior design and model specification.
  • Fitting, diagnostics, and criticism.
  • Sensitivity analysis.
  • Final communication.

Required outputs

  • reproducible code;
  • prior predictive checks;
  • posterior diagnostics;
  • predictive evaluation;
  • sensitivity analysis;
  • written discussion of limitations.

Computational lab

  • Use Projects as the canonical project bank.

Common pitfalls

  • Starting with a complicated model before articulating the scientific question.
  • Underreporting computational failures.
  • Omitting sensitivity analysis.

Exercises

  • Draft a one-page project proposal with model family, prior strategy, diagnostics plan, and evaluation criteria.

Full chapter

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References

Berger, James O. 1985. Statistical Decision Theory and Bayesian Analysis. 2nd ed. Springer.
Gabry, Jonah, Daniel Simpson, Aki Vehtari, Michael Betancourt, and Andrew Gelman. 2019. “Visualization in Bayesian Workflow.” Journal of the Royal Statistical Society: Series A 182 (2): 389–402.
Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. CRC Press.
MacKay, David J. C. 2003. Information Theory, Inference, and Learning Algorithms. Cambridge University Press.
Murphy, Kevin P. 2023. Probabilistic Machine Learning: Advanced Topics. MIT Press.
Robert, Christian P. 2007. The Bayesian Choice. 2nd ed. Springer.
Vehtari, Aki, Andrew Gelman, and Jonah Gabry. 2017. “Practical Bayesian Model Evaluation Using Leave-One-Out Cross-Validation and WAIC.” Statistics and Computing 27 (5): 1413–32.