Roughly speaking, Bayesian Workflow is to Bayesian Data Analysis in 2026 what Bayesian Data Analysis was to earlier Bayesian books in 1995: it builds upon everything that came before.

With Bayesian Data Analysis, the big steps forward were:

• Going beyond Bayesian inference to also consider Bayesian model building (as a researcher, you construct the model, it isn’t just given to you as in a textbook), model checking (breaking through the absolutely horrible attitude, common to Bayesians in the early 1990s, that the model was “subjective” and thus should not be checked), and model improvement (continuous model expansion, not the misguided idea of assigning posterior probabilities).
• Going beyond simple conjugate models. BDA had lots of hierarchical models, also lots of computational tools so that you could fit the models you want by putting them together from understandable components. And I like how we had a clear separation between modeling and computing. The model comes first, then you figure out how to compute it. Or you set up a model that works within your computational constraints.
• A Bayesian approach to sampling and causal inference. This was Rubin’s framework in which unobserved units in the population and unobserved causal outcomes are treated as missing data and are part of a joint probability model. We worked this out in chapter 7 of BDA (which became chapter 8 in the third edition of the book).
• Lots of live examples. Not just “real-data examples,” but problems we’d directly worked on. This motivated us and I think it gave our readers a sense of how Bayesian methods worked not just in theory but in applied problems.
• A pragmatic view of probability as a measurable quantity. That’s right there in chapter 1. Bayesian methods are not the product of a philosophical stance; they’re a way to connect models and data using probability.

I could go on and on, but for that I can refer you to the Bayesian Data Analysis book.

And these are the key innovations of Bayesian Workflow:

• Going beyond Bayesian data analysis (model building, inference, model checking, and model expansion) to consider the larger process of statistical modeling, including comparisons of multiple models fit to a single dataset.
• A fuller use of informative priors. This is a big deal. In BDA we still had a bit of the Bayesian cringe going on. One reason we’ve moved toward stronger priors is that the replication crisis has taught us that the amount of prior information available in any given problem is often approximately the same as the information coming from an experiment (see here, for example). Informative priors also fit our increased focus on generative modeling, and we’re doing a lot more prior predictive checking to understand the implications of our models.
• More integration between modeling, data analysis, and computing. One way to see this is that the Bayesian Workflow webpage has the code to run all our examples. We also have lots of code snippets in the text as a way of demonstrating the way in which coding is central to our statistical workflow.
• Lots more live examples. It’s been 30 years since BDA first came out. One reason that Bayesian Workflow has 11 authors is that different collaborators worked on different examples (but the three principal authors read through the entire book, so the general approach should remain coherent).
• Simulation-based experimentation. This is something my colleagues have been doing more and more over the years. At its most basic, simulation-based experimentation provides a best-case baseline for statistical methods: if you can’t recover your quantities of interest with sufficient accuracy under ideal conditions (when your data are simulated from the model you’re fitting), then you know you’re in trouble. And often this is the case! Beyond that, we can simulate from one model and fit another, and see what happens. Simulation experiments aren’t always so easy to construct, as they involve specifying the entire data-generation process. But we think this is effort worth expending, as it involves thinking about the problem you’re working on.

I could go on and on, but for that I can refer you to the Bayesian Workflow book.

And now for the reviews

But you don’t have to trust me on this! Just listen to some of the eminent statisticians and educators who’ve reviewed our book:

Bin Yu (University of California):

An outstanding, protocol-driven guide for Bayesian data analysis, Bayesian Workflow by Gelman, Vehtari, McElreath and co-authors delivers a practical and comprehensive framework for iterative modeling, emphasizing simulation, diagnostic checks, and rigorous empirical validation, and with a long and impressive list of case studies. By treating data analysis as a structured, verifiable workflow, it provides an indispensable toolkit for diagnosing model failures, refining priors, and building reliable data analysis systems for reproducible conclusions, useful for beginning and veteran data analysts alike.

David Spiegelhalter (Cambridge University):

This is not a typical methods textbook, but instead it guides the reader through the whole process of fitting, critiquing and adapting statistical models to real-world problems. It is full of the accumulated wisdom of skilled practitioners, teaching through demonstration rather than theory, with both basic and highly sophisticated examples. I strongly recommend this book to statisticians who really want to understand what they can learn from their data.

Brad Efron (Stanford University):

A bravura performance…Gelman, Vehtari, McElreath and friends develop in detail a practical Bayesian data analysis workflow, from acquisition to final report, including full computational guidance.

Christian Robert (Université Paris Dauphine):

This original, thought-provoking, and transformative book is much much more than an implementation manual for Bayesian Data Analysis, even though it shares almost the same perspective. (The first sentence of the book states that the authors’ “conceptions of statistical practice, and of Bayesian statistics, have changed over the years”.) By providing a modus vivendi for undertaking Bayesian modelling from scratch in realistic settings where models are not magicked out of the blue, the authors explicit and rationalise the many steps required by such a bottom-up modelling protocol (“not a checklist, not a cookbook”, and not a flowchart!) in real situations. The contents read very well and very smoothly, with a seamless conjunction of intuition, modelling advices, computational details, and comparison tools. While unsurprisingly Bayesian, the perspective adopted therein remains both open and inclusive, with a welcome humility about the limitations and challenges of Bayesian workflows. This book should thus appeal to and profit a wide variety of readers, as providing guidance through an extensive collection of highly detailed examples, with shared code and exercises.

Rohan Alexander (University of Toronto):

Some statistics books show you how to beat an egg, others are recipe books: if this, then that style. This book teaches you how to cook. Written by authors who established so much of how we do Bayesian statistics, this new book is an indispensable guide for analyzing data in a trustworthy way. It walks you through the actual steps involved in building models to explore and understand datasets. Part 4 is particularly excellent – the authors provide many end-to-end case studies that will be useful for both practitioners and students. It highlights the value of their workflow-based approach. Filled with chatty asides, the book introduces the Bayesian workflow to a broad audience. It embraces the frustrations and complexities of actually doing Bayesian statistics and provides specific guidance throughout. Each chapter contains exercises and it could be the basis of an upper-year undergraduate course, or a first-year grad course, in applied statistics. It will be used for many years to come.

Mine Doğucu (Harvard University):

What makes Bayesian Workflow so exceptional is how it seamlessly pairs profound ideas about modeling with the adoption of modern computational practice. By centering the messy, iterative process of modeling through real-world case studies, the authors reject rigid cookbooks and checklists in favor of building deep situational awareness. Because the ideas are so clearly articulated and deeply applied, this book serves as an invaluable pedagogical resource. With its practical exercises, individual chapters or the text as a whole can easily be integrated into upper-level undergraduate or graduate courses, while also remaining accessible for self-guided readers. It is an indispensable read for anyone with foundational knowledge in Bayesian methods, regardless of whether they are applied practitioners, software developers, or methodologists.

You might also be interested in the journal issue on statistical workflow that we recently edited for the Philosophical Transactions of the Royal Society.

Again, here’s Bayesian Workflow on Amazon, here’s the publisher’s website, and here’s our website with data, code, and lots more.

Enjoy.

This post is by Bob.

I’ve been thinking a lot lately about R-hat given that I’m using it for online converging monitoring in our new Walnuts implementation. In that setting, where I use Welford accumulators to update R-hat estimates every iteration, I can’t use split R-hat without way too much buffering. So I’ve been thinking about the effect of splitting, too, and whether we need it. I asked Andrew and he said Kenny Shirley once produced an example where split R-hat diagnosed non-convergence that regular R-hat didn’t, but that example is lost to time and we’ve never seen this kind of behavior with NUTS as far as I know (please give us an example in the comments or via email to Andrew if you have).

Relating R-hat and ESS

My intuition was that we could set a low enough R-hat threshold that it would ensure a high enough effective sample size (ESS) when we crossed it. The relation’s a little tighter than I thought, with

    Rhat^2 ≈ 1 + M / ESS,

where M is the number of chains and ESS is the effective sample size of all chains combined. There’s a multivariate proof in Vats and Knudson, 2021, Revisitng the Gelman-Rubin diagnostic, Statistical Science, page 2 and section 5 for details, but it’s pretty straightforward to get the intuition when you reduce R-hat^2 to (N-1)/N + var(chain-means) / man(chain-variances) as Charles Margossian did in his nested R-hat paper. Vats and Knudson disapprove of Andrew and Aki’s suggested threshold of 1.1 from BDA3, because it is satisfied with a combined ESS of 20 across Andrew’s default 4 chains.

Being me, I tried to validate my intuition with simulations rather than linear algebra. Also, I like to see that things work in practice that theory entails to make sure I’ve understood all the assumptions baked into the theory (one can’t prove anything without assumptions!). When asked to code a simulation using ArviZ, Claude inserted a (2 * M) in the numerator in place of the M. Where did that come from, I asked? It told me it needed the factor of 2 because ArviZ uses split Rhat. D’oh! Of course it does, because we’ve doubled M without increasing ESS.

A worked example

Suppose we have 4 chains with a combined ESS of 400. Then sqrt(1 + 4/400) ≈ 1.005 and sqrt(1 + (2 * 4) / 400) ≈ 1.01. We’ve effectively doubled the number after the 1 by splitting. Unlike Vats and Knudson, I usually don’t need an ESS >> 100, so the 400 required for split R-hat < 1.01 is perhaps a bit too conservative for my tastes. On the other hand, we face a practical problem estimating ESS reliably with fewer than 50 or so ESS per chain. Estimation is challenging because it relies on autocorrelation estimates from the chains themselves, which become much noisier when based on shorter chains. (Side question: Do we not combine autocorrelation estimates across chains to reduce standard error because some chains might not be mixing?) Also, we know this algebra wasn't a coincidence of 4 chains and 400 draws. The Taylor expansion of sqrt(1 + x) is the convergent sequence

    sqrt(1 + x) = 1 + x/2 - x^2 / 8 + x^3 / 16 + ...

When x < 0.1, the first-order approximation, sqrt(1 + x) = 1 + x / 2, is good.

The bottom line for practitioners

We need around twice as many draws to get below a fixed threshold with split R-hat than with the original R-hat.

This post is from Bob.

I’ve been thinking a lot about scores (gradients of the log density function) and how they can be used for convergence monitoring. We know that the expected value of the score is zero. Stein generalized this with Stein operators. In the monomial case, the Stein operators give you functions in increasing degrees, all of which have zero expectation in the posterior. Here theta is the variable being sampled and S is the score function, so that S(theta) is the gradient of the target log density evaluated at theta.

    Order 0: S(theta)

    Order 1: 1 + theta .* S(theta)

    Order 2: 2 * theta + theta^2 .* S(theta)

This leads to a natural test for convergence of first, second, and third moments. Just compute Monte Carlo estimates of these quantities and see if they’re zero. We’d want to standardize for standard deviation to make the result scale-free like R-hat. To develop some intuitions, in a standard normal distribution p(theta) = normal(theta | 0, I), we have S(theta) = -theta, and thus S(theta) converges to zero at the same rate as our variable theta converges to its true value; the order 1 test is 1 – theta^2, which we know has expectation zero because theta^2 has a ChiSquared(1) distribution with expectation of 1). The order 1 case corresponds to equipartition in physics and the form D + theta’ * S(theta) also naturally has zero expectation as shown in the viral theorem in physics in the 1870s.

Diving into this a bit more led me back to Jackson Gorham and Lester Mackey’s work on Stein’s method. They haven’t been sitting still since introducing the basic idea, which kernelizes the idea above. Mackey et al. have produced an absolutely wonderful summary of this body of work in two forms. The first is a dense, 41-slide deck with all the key definitions and results. I’d suggest at least skimming this first.

Lester Mackey. April 2026. Stein’s Method, Learning, and Inference.. GitHub.

Mackey along with Chris Oates and Qiang Liu, who have also worked heavily in this area, put together a definitive monograph. They’ve presented a great deal of difficult material in a way that I can digest (though it’s going to be rough going if you’re not well versed in sampling and how MCMC is traditionally measured and evaluated).

Qiang Liu, Lester Mackey, Chris Oates. March 2026. Probabilistic Inference and Learning with Stein’s Method. arXiv.

In particular, they go over Stein variational inference, which seems to me like it would be the ideal way to perform quasi Monte Carlo-like inference for statistical models if we could only get a robust version to scale. The idea’s to initialize a bunch of points, then use optimization to minimize a kernelized Stein discrepancy of the empirical distribution of those points to the true distribution.

David Eubanks writes:

My site is kappazoo.com, and it’s still a work in progress. I would rather have emailed after I had the new goodness-of-fit code done, but I saw that you’re doing a summer workshop (on Andrew’s blog) [editor: Modern Modeling Methods (M3)], so thought I’d mention it now.

It may be billed as a work in progress, but it’s a complete draft with no missing sections that provides a really nice overview of rating/crowdsourcing models. These are the models that dragged me into statistics, namely Bayesian rating models formulated as noisy measurement models. The first model of this kind that I or Eubanks could find was Phil Dawid and Allan Skene’s (1979) paper on rating.

Eubanks works through a great deal of workflow without calling it that. There are multiple model evaluation and comparison measures used and explained with connections to information-theoretic notions like entropy.

There’s a long discussion of Cohen’s kappa statistic, which is a commonly reported statistic measuring inter-rater agreement. As Eubanks notes, it doesn’t deliver on its promise of adequately measuring inter-rater agreement. The discussion is quite good here and complementary to the discussion from me and Becky Passonneau in our paper on rating in NLP, though our conclusions are the same.

I was surprised to see that Eubanks has a section comparing item-response theory (IRT) models with difficulty. I’ve been trying to convince people this is important for years. It took me around ten years to figure out how to move from Dawid and Skene’s IRT-0-like model to an IRT-1-like model, which we report in in our latest paper on crowdsourcing with difficulty parameters (which also works through a lot of Bayesian workflow in considering different models). I can’t identify what took so long—it seems so obvious to me now.

We’re very excited about this book. It’s the result of several years of effort. You can pre-order from the publisher or from Amazon.

Here’s the book’s webpage, which includes the data and code for the book’s examples and case studies, of which there are many.

Here’s the table of contents:

Part 1: From Bayesian inference to Bayesian workflow
1. Bayesian theory and Bayesian practice
2. Statistical modeling and workflow
3. Computational tools
4. Introduction to workflow: Modeling performance on a multiple choice exam

Part 2: Statistical workflow
5. Building statistical models
6. Using simulations to capture uncertainty
7. Prediction, generalization, and causal inference
8. Visualizing and checking fitted models
9. Comparing and improving models
10. Statistical inference and scientific inference

Part 3: Computational workflow
11. Fitting statistical models
12. Diagnosing and fixing problems with fitting
13. Approximate algorithms and approximate models
14. Simulation-based calibration checking
15. Statistical modeling as software development

Part 4. Case studies
16. Coding a series of models: Simulated data of movie ratings
17. Prior specification for regression models: Reanalysis of a sleep study
18. Predictive model checking and comparison: Clinical trial
19. Building up to a hierarchical model: Coronavirus testing
20. Using a fitted model for decision analysis: Classification competition
21. Posterior predictive checking: Stochastic learning in dogs
22. Incremental development and testing: Black cat adoptions
23. Debugging a model: World Cup football
24. Leave-one-out cross validation model checking and comparison: Roaches
25. Model building and expansion: Golf putting
26. Model building with latent variables: Markov models for animal movement
27. Model building: Time-series decomposition for birthdays
28. Models for regression coefficients and variable selection: Student grades
29. Sampling problems with latent variables: No vehicles in the park
30. Challenge of multimodality: Differential equation for planetary motion
31. Simulation-based calibration checking in model development workflow

Appendices
A. Statistical and computational workflow for Bayesians and non-Bayesians
B. How to get the most out of Bayesian Data Analysis

One way to think of the book is that it’s all the things missing from BDA, like how to set up an informative prior, what to do when your computations aren’t converging, how to work through a series of models fit to the same data, how to design and perform simulated-data experiments . . . and all sorts of other things too.

The core of the book–parts 1 through 3–clock in under 200 pages, and then we have another 300 pages full of case studies demonstrating different aspects of Bayesian statistical and computational workflow. The appendices should be useful to you too, first because the workflow ideas in this book apply to non-Bayesian inference too, and second because BDA still has lots of valuable material in it, so it’s good to know where to look.

This new Bayesian Workflow book could change your life (we hope), and I thank my coauthors, Aki Vehtari and Richard McElreath, with Daniel Simpson, Charles C. Margossian, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, Martin Modrák, Vianey Leos Barajas, for all their care and effort. We thank our employers and various funding agencies for giving us the resources to be able to write this book as a side project along with all our daily responsibilities. And we thank many people for their input on earlier versions of the book, along with the Stan developers making so much of this work possible and the Stan community of users for supplying a continuing series of challenges that have motivated many of the ideas and methods discussed in the book.

I hope you find the book readable, interesting, and useful.