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Bayesian Workflow

Andrew Gelman, Aki Vehtari, Daniel Simpson, Charles C. Margossian, Bob Carpenter, Yuling Yao, Lauren Kennedy, Jonah Gabry, Paul-Christian Bürkner, Martin Modrák

arXiv:2011.01808v1stat.ME

TL;DR

Applied Bayesian analysis requires more than inference alone: practitioners must manage model construction, checking, computation, comparison, and reproducibility. This review brings these activities together through examples and presents Bayesian workflow as a unified framework, while identifying the need for more structured guidance and worked examples.

  • Problem

    Bayesian methods need to be understood in practical use, including the broader workflow of applied data analysis beyond Bayesian inference.

  • Method

    The review examines Bayesian workflow through fake-data simulation, predictive checking, model stacking, reproducibility practices, and computational diagnostics across worked examples.

  • Results

    The paper provides a first step toward placing the many activities of Bayesian workflow under a single roof.

  • Takeaways & Limitations

    Bayesian workflow encompasses iterative model evaluation, computational troubleshooting, model comparison, and reproducibility alongside inference.

  • Takeaways & Limitations

    The review still needs more structure, worked examples with code, and clearer paths for conducting Bayesian analyses.

Abstract

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The Bayesian approach to data analysis provides a powerful way to handle uncertainty in all observations, model parameters, and model structure using probability theory. Probabilistic programming languages make it easier to specify and fit Bayesian models, but this still leaves us with many options regarding constructing, evaluating, and using these models, along with many remaining challenges in computation. Using Bayesian inference to solve real-world problems requires not only statistical skills, subject matter knowledge, and programming, but also awareness of the decisions made in the process of data analysis. All of these aspects can be understood as part of a tangled workflow of applied Bayesian statistics. Beyond inference, the workflow also includes iterative model building, model checking, validation and troubleshooting of computational problems, model understanding, and model comparison. We review all these aspects of workflow in the context of several examples, keeping in mind that in practice we will be fitting many models for any given problem, even if only a subset of them will ultimately be relevant for our conclusions.

1. Introduction

Bayesian workflow extends Bayesian inference into an iterative process of model building, computation, checking, improvement, understanding, and comparison. The paper reviews this tangled workflow and its progression from examples toward more systematic practice.

  • Bayesian workflow includes model building, inference, model checking and improvement, and comparison of different models.
  • Analysts commonly fit a series of models, including poor choices, useful but flawed models, and models ultimately worth reporting.
  • Workflow is needed because computation is challenging, model choice is often uncertain, and related models can clarify fitted-model understanding.
  • Different models can yield different conclusions without one being clearly preferable, making multiple models useful for displaying model-choice uncertainty.
  • The paper treats workflow as more general than an example but less precisely specified than a method, with methodology becoming increasingly codified from example to theory.
  • The reviewed workflow is more tangled than textbook workflows because complex computation and many nested sub-workflows require experimentation and checking.
  • Practical constraints can force shortcuts, which the paper frames as approximations to the full workflow for more informed allocation of limited time and effort.

2. Before fitting a model

Before fitting, Bayesian workflow develops models iteratively, chooses interpretable parameterizations, and uses prior predictive simulation to examine model implications. Generative-model thinking also clarifies what the observations can and cannot identify.

  • Model construction: Analyses may begin with a simple model, a large model to simplify, or multiple distinct approaches to the same data.
  • Model construction: Model construction is modular and iterative, allowing distributions, functions, measurement-error components, and priors to be replaced as needed.
  • Scaling and transforming parameters: Interpretable, naturally scaled parameters facilitate informative priors and effective hierarchical modeling.
  • Prior predictive checking: 1000 simulations of logistic-regression coefficients and outcomes show that weak independent priors can imply predictions concentrated near ȳ = 0 or 1 when covariates increase.
  • Prior predictive checking: Prior predictive checks simulate data from the model before using observed data, helping reveal implications of priors and refine models without reusing the data.
  • Prior predictive checking: Joint priors can control the overall complexity of larger parameter sets and produce more sensible prior predictions than independent priors.
  • Generative and partially generative models: A generative model is required for predictive simulation and model checking, although Bayesian inference itself only requires the likelihood.
  • Generative and partially generative models: Generative-model assumptions can improve inferential efficiency but can also be wrong, motivating workflow checks.

3. Fitting a model

Fitting a Bayesian model requires choosing among computational methods, diagnosing reliability, and adapting effort to the model’s place in the workflow. Fast approximations help explore many models, while MCMC is generally needed for accurate final posterior exploration.

  • Algorithms and diagnostics: The paper focuses on Hamiltonian Monte Carlo and its variants in Stan, while noting that similar principles apply with algorithm-specific differences elsewhere.
  • Algorithms and diagnostics: Reliable Bayesian workflow requires inference algorithms with strong diagnostics, and the paper discusses such diagnostics for HMC.
  • Algorithms and diagnostics: Warmup reduces dependence on initial values, tunes algorithmic parameters, and can quickly flag computationally problematic models during exploration.
  • Algorithms and diagnostics: Standard practice is to continue until bR is less than 1.01 for all parameters and quantities of interest, while also monitoring multivariate mixing.
  • Algorithms and diagnostics: MCMC theoretically reduces approximation error with more simulations, and mixed chains support estimation of Monte Carlo standard error.
  • Choosing computational effort: MCMC is not always scalable for large data and models, creating a speed–accuracy tradeoff with approximate algorithms.
  • Choosing computational effort: Early workflow stages can use approximate methods for large-scale posterior features, whereas late stages usually require MCMC for delicate features.
  • Choosing computational effort: There is no one-size-fits-all approximate inference algorithm, although established model components can support construction of appropriate approximations.

4. Using constructed data to find and understand problems

Constructed-data simulations validate computation and reveal what a model can learn under controlled conditions, while exposing weaknesses in data informativeness, model specification, and inference. They also show how treatment-assignment balance and functional-form assumptions affect estimated effects.

  • Validation with constructed data: Simulated data isolate computational and modeling problems from awkward real-data fit issues, enabling direct checks against known parameters.The same workflow can assess data models, priors, and applied inference methods.
  • Validation with constructed data: Fake-data recovery tests whether procedures reproduce known parameter values and whether posterior intervals provide appropriate uncertainty coverage.Point estimates and posterior-interval coverage can both be examined.
  • Troubleshooting: A failed fake-data check warrants simplifying the model until it works, then locating the component responsible for the failure.This helps distinguish uninformative model components from broader computational or specification problems.
  • Interpreting what models can learn: Fake-data simulations provide an upper bound on latent-process learning: success on model-generated data does not guarantee sensible real-data inference, but failure makes it unreasonable to expect good real-data inference.Even good observed-data fit may be insufficient for reliable latent-variable conclusions.
  • Simulation-based calibration: Single truth-point simulations can mislead because calibration generally holds over the prior, while noninformative data or random variation can prevent recovery at one parameter value.For central values under approximately unimodal prior and posterior distributions, posterior intervals may overcover.
  • Simulation-based calibration: Simulation-based calibration is more comprehensive than truth-point benchmarking but requires repeated model fits and substantial computation, especially without parallelization.The authors describe benchmarking and calibration as a spectrum from detecting gross problems to identifying finer computational issues.
  • Treatment-assignment experiments: Under unbalanced assignment, the group difference was −13.8 ± 1.5 despite a true effect of 10, whereas regression adjustment recovered 9.7 ± 0.8.The simulation preferentially assigned treatment to less well-performing students.

5. Addressing computational problems

Computational difficulties often signal problems with the model, data informativeness, or posterior geometry rather than being isolated software issues. The recommended workflow alternates simplification, incremental construction, targeted simulation, and practical fitting checks.

  • Diagnosing computational problems: Poor convergence can correspond to substantively irrelevant or nonsensical parameter regions, although some models are legitimately difficult to fit.Code bugs and inappropriate model choices are among the possible underlying problems.
  • Debugging strategy: Debugging proceeds from both directions: simplify the failing model and incrementally add features to a simple, well-understood model until the problem appears.Separate components should also be fit independently on simulated data as unit tests.
  • Scope of model fitting: A complex model may remain unfitted because algorithms cannot handle it, data and priors are insufficiently informative, or exploration redirects the analysis.The initially intended model is therefore provisional rather than guaranteed.
  • Sources of slowness: Hamiltonian Monte Carlo can be slow because of expensive gradients, high dimensionality, or posterior geometry that varies across regions.Slow computation often indicates a poorly performing HMC fit, though not always a model defect.
  • Practical workflow: For a large multilevel logistic regression that failed to finish, the proposed workflow starts with simulated data and a smaller model before increasing iterations or complexity.The sequence included removing varying intercepts, then adding batches incrementally.
  • Practical workflow: Initial fitting should use short runs, moderately informative priors, and data subsets while checking whether the model serves prediction or process learning.The advice suggests 200 iterations initially, increasing to 2000 only after diagnosing the fit.

5.4. Monitoring intermediate quantities

Monitoring intermediate quantities makes computational failures more interpretable by connecting MCMC behavior to model predictions and internal calculations.

  • Visualization: Plotting saved intermediate quantities alongside MCMC output often reveals more than console print statements.bayesplot and ArviZ are cited as tools for these visualizations.
  • Diagnosing stuck chains: When chains remain in low-density regions, inspecting predictions at those parameter values can clarify why the sampler behaves incorrectly.The predictive view helps diagnose what is going wrong in otherwise baffling stuck-chain behavior.

5.5. Stacking to reweight poorly mixing chains

Poor mixing can reflect weakly identified or multimodal posterior geometry, and stacking offers an approximate way to downweight chains trapped in low-probability regions. Its usefulness depends on the geometry and predictive interpretation of the modes.

  • Diagnosing mixing problems: MCMC can move to unreasonable parameter regions because of model misspecification, weakly informative observations, or difficult posterior geometry.Other fits may mix adequately, so the observed failure pattern matters diagnostically.
  • Stacking: When chains mix slowly but remain broadly reasonable, stacking uses cross-validation weights to approximately discard chains stuck in low-probability modes.The resulting combination is not necessarily equivalent to a single posterior fit.
  • Posterior geometries: Multimodal problems include negligible minor modes, symmetric label-switching modes, and distinct high-mass modes with different substantive interpretations.The paper distinguishes these cases because they call for different remedies.
  • Posterior geometries: For negligible modes, initialization, added prior information, hard constraints, or approximate mode-mass estimation can help avoid or prune them.These remedies target effectively disjoint posterior volumes where all but one mode have near-zero mass.
  • Posterior geometries: Symmetric label-switching modes are commonly handled by restricting the model to identify the mode of interest.This is presented as standard practice for that specific geometry.
  • Posterior geometries: Stacking can approximately address distinct high-mass modes through predictive model averaging, but it is not completely general because it requires a predictive target.Such modes can represent competing regulatory regimes with opposite effect signs.
  • Posterior geometries: Arithmetically unstable tails may be tolerable when initialized near the main posterior mass, but rare-event questions require reparameterization.The paper notes limits to what default effective sample sizes of a few hundred to a few thousand can learn.

5.7. Reparameterization

HMC-based sampling works best with well-behaved posterior geometry, but even modest model complexity can create computational difficulties. Reparameterization can resolve funnel pathologies in hierarchical models.

  • HMC sampling is most effective when the joint posterior has no sharp corners, cusps, or other irregularities.Appropriate mass-matrix tuning also contributes to sampler performance.
  • Hierarchical models can develop funnel pathologies when group-level variance parameters approach zero.
  • Reparameterization can resolve computational difficulties caused by hierarchical-model funnel pathologies.

5.8. Marginalization

Marginalization improves sampling by drawing efficiently from a well-behaved marginal posterior and recovering other parameters conditionally. When exact densities are unavailable, Laplace approximations can support this strategy, sometimes with bias.

  • Funnel geometry can arise from interactions between a group-level scale parameter φ and an individual-level mean θ, even when φ’s marginal density is well behaved.
  • MCMC can sample the marginal likelihood p(y|φ) and then recover θ from the conditional distribution p(θ|φ, y) at low computational cost.
  • Laplace approximations can approximate p(y|φ) and p(θ|φ, y) when exact densities are unavailable, though coupling with HMC can introduce bias.

5.9. Adding prior information

Adding prior information can stabilize computation, but its usefulness depends on parameter scaling, data informativeness, and substantive knowledge. The declining-exponential example shows that a true model may still be poorly identified by the observed data.

  • Adding prior information: Reasonable prior information can fix many computation problems, although eliciting it from domain experts can be costly.
  • Adding prior information: The amount of information learned depends on the specific observed data realization, so classical identifiability labels can be misleading.
  • Adding prior information: With decay rates b1 = 0.1 and b2 = 2.0, simulated data allowed smooth fitting and recovery of all five model parameters.
  • Adding prior information: When the decay rates changed to b1 = 0.1 and b2 = 0.2, the exponentials became difficult to distinguish and Stan exhibited terrible convergence.
  • Adding prior information: Default normal(0, 1) priors can regularize parameters on roughly unit scales, with posterior sensitivity checked against alternative prior choices.
  • Adding prior information: Informative priors can accelerate MCMC mixing by increasing tail-log-concavity without necessarily introducing a model-bias versus computational-efficiency tradeoff.
  • Adding prior information: The computational-troubleshooting ladder links poor MCMC mixing to difficult geometry, weakly informative data, and substantive prior information as a solution.
  • Adding prior information: Stronger regularization can improve mixing, but haphazard model changes may blur target features and threaten inference validity.

5.10. Adding data

Additional data sources can constrain models within the Bayesian analysis. In small-data regimes, however, the full model may remain uninformed and require replacement by a simpler model.

  • New data sources, such as calibration experiments, can be incorporated within the model to inform quantities like response standard deviation.
  • Small datasets can produce funnel-like posterior geometry that is difficult to avoid, making a simpler model necessary when the full model is not data-informed.

6. Evaluating and using a fitted model

Evaluating a fitted model requires checks tailored to its goals, including posterior predictive checks, cross-validation diagnostics, and influence analysis. These tools reveal misfit, calibration problems, difficult observations, and influential data points that can guide model improvement.

  • Posterior predictive checking: Posterior predictive checks compare observed data or summaries with simulations from the posterior predictive distribution to detect model misfit.Checks may examine the full data distribution, summary statistics, or subgroups, especially groupings omitted from the model.
  • Posterior predictive checking: Posterior predictive checks can use tailored visualizations to expose patterns in observed data that replicated datasets fail to reproduce.The examples include density, statistic, bars, and subgroup checks, with subgroup deviations indicating uncaptured variability.
  • Cross-validation: Cross-validation improves predictive diagnostics by evaluating predictions on held-out data, particularly for flexible models.Unlike posterior predictive checking, cross-validation separates model fitting from misfit evaluation.
  • Cross-validation: LOO-CV diagnostics assess calibration, difficult observations, and observation influence, with importance-sampling approximations avoiding repeated refits.LOO-PIT values should be uniform under perfect calibration; concentration near the middle indicates overdispersed predictive distributions.
  • Cross-validation: 16.4 estimated difference in expected log predictive densities, with standard error 4.4, identifies 10–15 high-arsenic non-switchers as problematic observations.Pointwise log-score comparisons both quantify model differences and locate observations contributing large negative predictive scores.
  • Influence analysis: Influence analysis can reveal how parameter inferences change when observations are omitted, but clustered data may require removing multiple points.This approach remains computationally inexpensive and useful in many applied settings.

7. Modifying a model

Modifying Bayesian models is an iterative process driven by new data, model failures, computational constraints, domain knowledge, and prior sensitivity. Model expansion and prior adjustment must account for how added parameters alter the joint model and its implied predictions.

  • Model expansion: Model building combines reusable statistical components to represent new data, additional features, and links to underlying processes.Components may include linear, logistic, and exponential functions; common probability distributions; and varying coefficients.
  • Model expansion: Model expansion can respond to new data, failures in existing predictions, or computational struggles with current fitting procedures.The election-forecasting example began from an earlier poll-aggregation model after prediction failures in certain swing states.
  • Data models: Data models should combine fit to observed data with domain expertise, while accounting for how data connect to underlying parameters.Preprocessing and dimensionality reduction can make generative models approximate, with data quality and bias remaining important concerns.
  • Data models: Combining measurements of different quality may require analytic approximations that incorporate aggregated data into the likelihood without modeling every latent outcome.One pharmacological example combines individual-level data for one drug with average data for a competitor’s product.
  • Prior distributions: Priors act as model constraints, so stronger priors on nonlinear components can keep an estimated curve close to linear.The prior may represent substantive information or simply be part of the model specification.
  • Prior distributions: As models become more complex, priors often need tightening because parameter roles, hierarchical depth, and dimensionality affect the required prior information.With 26 regression predictors, independent weak priors can jointly favor high Bayesian R2 values; scaling priors with dimension can address this concentration.
  • A topology of models: Model topology links neighboring models and parameters, while prior distributions add a continuous dimension that bridges models.Search systems such as the Automatic Statistician use inference and model criticism to explore model and data spaces.

8. Understanding and comparing multiple models

Bayesian workflow fits multiple models to understand how modeling choices affect inferences, not merely to select or average models. Comparisons, multiverse analyses, and stacking help characterize uncertainty across alternatives and combine predictive information when appropriate.

  • Visualizing models: Bayesian workflow fits many models for one problem to explore model-fitting decisions and understand how conclusions change across alternatives.The motivations include building from simple models, correcting mistakes, incorporating new data, changing priors, and responding to computational problems.
  • Visualizing models: A sequence from simple to increasingly adjusted models shows how an inference changes as modeling choices are added.The goal is understanding the path to a final model, not necessarily selecting or averaging models.
  • Visualizing models: Keeping earlier models as scaffolds preserves useful comparisons after a model is checked, expanded, or replaced.Simple unadjusted models can show what later causal adjustments have changed.
  • Visualizing models: Model comparisons can reveal researcher degrees of freedom because fitted models may not represent all plausible alternatives.Selecting a single best model can also create overfitting concerns.
  • Multiverse analysis: Multiverse analysis uses multiple models and analytical choices to assess how conclusions vary, reducing the importance of identifying one best model when conclusions are stable.Choices may include preprocessing, response distributions, and evaluation metrics.
  • Cross-validation and model averaging: Cross-validation compares models on shared data, while stacking combines joint predictions and can help when many similar weak models are available.When comparison uncertainty is non-negligible, selecting only the model with the best cross-validation score is discouraged.
  • Cross-validation and model averaging: Stacking can behave like pointwise model selection, with weights reflecting how often competing models outperform one another.The paper presents stacking as a possible step toward modeling heterogeneity rather than an endpoint.
  • Cross-validation and model averaging: Candidate models may include deliberately simple, experimental, flawed, or coding-error models that are useful for workflow but not for averaging.Variable selection seeks a smaller model with predictive performance comparable to an expanded model when many candidates exist.

9. Modeling as software development

Developing Bayesian models is software development involving model code, data preprocessing, computation, testing, documentation, and reproducibility. Modular design, version control, scripts, and explicit testing help manage complexity and preserve reliable analyses.

  • Software development: Probabilistic model development includes writing and debugging code, preprocessing data, and later understanding, communicating, and using inferences.The many synchronized moving parts make software development practices relevant to Bayesian workflow.
  • Version control: Version control should cover code, reports, graphs, and data so projects can be reverted, synchronized, and packaged at publication milestones.Public repositories can increase transparency for policy models by documenting the model, data, inference parameters, and scripts used.
  • Design and testing: Top-down design starts from user goals and specifies data inputs, the probabilistic model, priors, simulators, and model tests.Bottom-up development builds larger functionality from well-tested foundational components.
  • Design and testing: Modularity makes computational code easier to document, read, debug, maintain, and modify by composing larger functions from trusted smaller pieces.Repeated code should be encapsulated in reusable functions.
  • Design and testing: Standardization functions require explicit choices about the definition of sd(v), including whether the denominator is n or n −1.Unresolved choices can create biases during inference, especially for vectors of length 1.
  • Design and testing: Input-output checks should flag illegal inputs early rather than allowing errors to propagate into later divide-by-zero failures.Higher-level numerical functions also require testing for boundary conditions, instability, imprecision, and derivative behavior.
  • Reproducibility: Reproducibility aims to let another person recreate an essentially equivalent analysis and report, although bit-level reproduction may be impossible.If reproduction changes the outcome, the original results are described as not particularly robust.
  • Reproducibility: Self-contained scripts provide executable documentation and can be more practical than one large notebook for complex projects.Readable code also depends on consistent naming, organization, and avoiding opaque inline comments.

10. Example of workflow involving model building and expansion: Golf putting

The golf example develops a Bayesian workflow by starting with simple models, comparing them with data, and expanding them when new data expose systematic failures. A geometry-based model improves on logistic regression, while later additions address distance uncertainty and residual model error.

  • Initial model building: The workflow begins by modeling putting success as a function of distance, first using logistic regression and then a geometry-based model.The geometry model represents the angular range that sends a ball into the hole and models shot-angle error with a normal distribution.
  • Initial model building: The geometry-based model fits the initial golf data much better than logistic regression while using one fewer parameter.Its sole unknown parameter is the standard deviation of shot angles, estimated as σ̂ = 1.53° with standard error 0.02.
  • Model checking: New putting data show higher success rates for short putts and lower-than-predicted success beyond 20 feet, revealing that the earlier model needs improvement.The higher short-putt success may reflect measurement differences or improved golfers, whereas the long-distance discrepancy is a systematic model failure.
  • Model expansion: The expanded model adds uncertainty in distance as well as angle, but the two-parameter version remains a gross simplification and uses average rather than exact shot distances.The model introduces σangle and σdistance to account for these sources of uncertainty.
  • Model expansion: The binomial likelihood overweights high-count short-distance observations, producing a problematic overall fit and poor MCMC convergence when the model cannot fit all data simultaneously.The fitted curve hugs the lowest-distance observations, while chains struggle between parameter values fitting those points and the bulk of the data.
  • Model checking and refinement: Adding a flexible error term yields a good fit: σangle = 1.02°, σdistance = 0.08, and σy = 0.003, with residuals showing no strong pattern.The model predicts success rates within at most half a percentage point at most distances, leaving no clear development direction from the current data.

11. Example of workflow for a model with unexpected multimodality: Planetary motion

The planetary-motion case study shows how a Bayesian workflow diagnoses unexpected multimodality through mechanistic modeling, simulation, diagnostics, and domain reasoning. Multiple chains exposed locally attractive, poorly fitting modes with negligible probability mass, motivating informed computational adjustments rather than indiscriminate chain removal.

  • 11.1. Mechanistic model of motion: A mechanistic model based on Newton’s laws represents planetary motion with coupled first-order differential equations and supports inference about physical quantities such as stellar mass.The model tracks position and momentum over time while incorporating gravitational interaction and domain knowledge.
  • 11.2. Fitting a simplified model: The initial Stan fit of the complete model failed dramatically: chains did not converge and required long runtimes, prompting a simpler model and simulated-data checks.The simplified model fixed most parameters, estimated k, and used quadrature alongside MCMC to understand the computational problem.
  • 11.3. Understanding the modes: The posterior’s tail modes arise from the periodic data structure, fit observations poorly, and contribute negligible probability mass, whereas the dominating mode contains essentially the relevant mass.Higher k values also made Hamilton’s equations harder to solve because stronger gravitational interaction increases the planet’s speed.
  • 11.4. Building up the model: Posterior predictive checks and conditional-likelihood analysis connected local modes to the star’s position and guided starting points toward the dominating mode in the complete model.The conditional likelihood varied q∗ while holding other parameters fixed, exposing suspected modes that supported the physical interpretation.
  • 11.5. General lessons from the planetary motion example: Starting points should be adjusted iteratively when defaults conflict with substantive expertise, but misbehaving chains should not be discarded before their causes and probability mass are analyzed.The authors distinguish computationally irrelevant local modes from modes that could signal flaws in the model or inference.

12. Discussion

Bayesian workflow treats applied modeling as iterative model navigation rather than a one-shot inference problem. It integrates model exploration, computational troubleshooting, model checking, comparison, adaptation to data and questions, and extensions toward new real-world settings.

  • Model exploration: Applied statistical work often explores a series of models, including models with poor fit, weak prediction, or slow convergence, before arriving at useful conclusions.This model-exploration perspective differs from traditional inference and machine-learning workflows, which prioritize posterior computation or predictive accuracy, respectively.
  • Model navigation: Model navigation is proposed as a broader data-science step that facilitates exploration of model space beyond isolated case-study practices.The paper places this proposal after historical advances in data summarization, modeling, and computation.
  • Iterative model building: Iterative model building helps researchers learn gradually, understand the modeled phenomenon, and manage the human and computational costs of fitting richer models.Starting from simpler models can make modeling more efficient in human time, while model iteration also addresses finite computational resources and failures of fully automated computation.
  • Model selection and overfitting: Data-dependent iterative improvement can create double dipping and post-selection concerns, so the workflow emphasizes understanding and justifying each modeling decision rather than selecting the best fit.The authors argue that their approach avoids the worst overfitting problems by reassessing choices in response to model problems and incorporating additional information where possible.
  • Model checking: Severe testing of model assumptions is presented as a basis for greater trustworthiness, even when the model resulted from data-dependent iterative workflow.The comparison is with a preregistered model whose assumptions have not been tested.
  • Applications and future directions: Bayesian workflow adapts models to the application and available data, extends beyond fitted-model inference to predictions for new settings, and aims to strengthen confidence in analyses and decisions.The broader framework can include different measurement modes, treatment assignments, production use, feedback, automation, and domain expertise.
  • Going forward: The paper is a review and first step toward organizing Bayesian workflow, while acknowledging the need for worked examples, code, and more structured guidance for practitioners.The authors point to future resources, including guidance and a book using Stan.
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