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.

Introduction
- From Bayesian inference to Bayesian workflow
- Why do we need a Bayesian workflow?
- “Workflow” and its relation to statistical theory and practice
- Organizing the many aspects of Bayesian workflow
- Aim and structure of this article
Before fitting a model
- Choosing an initial model
- Modular construction
- Scaling and transforming the parameters
- Prior predictive checking
- Generative and partially generative models
Fitting a model
- Initial values, adaptation, and warmup
- How long to run an iterative algorithm
- Approximate algorithms and approximate models
- Fit fast, fail fast
Using constructed data to find and understand problems
- Fake-data simulation
- Simulation-based calibration
- Experimentation using constructed data
Addressing computational problems
- The folk theorem of statistical computing
- Starting at simple and complex models and meeting in the middle
- Getting a handle on models that take a long time to fit
- Monitoring intermediate quantities
- Stacking to reweight poorly mixing chains
- Posterior distributions with multimodality and other difficult geometry
- Reparameterization
- Marginalization
- Adding prior information
- Adding data
Evaluating and using a fitted model
- Posterior predictive checking
- Cross validation and influence of individual data points and subsets of the data
- Influence of prior information
- Summarizing inference and propagating uncertainty
Modifying a model
- Constructing a model for the data
- Incorporating additional data
- Working with prior distributions
- A topology of models
Understanding and comparing multiple models
- Visualizing models in relation to each other
- Cross validation and model averaging
- Comparing a large number of models
Modeling as software development
- Version control smooths collaborations with others and with your past self
- Testing as you go
- Making it essentially reproducible
- Making it readable and maintainable
Example of workflow involving model building and expansion: Golf putting
- First model: logistic regression
- Modeling from first principles
- Testing the fitted model on new data
- A new model accounting for how hard the ball is hit
- Expanding the model by including a fudge factor
- General lessons from the golf example
Example of workflow for a model with unexpected multimodality: Planetary motion
- Mechanistic model of motion
- Fitting a simplified model
- Bad Markov chain, slow Markov chain?
- Building up the model
- General lessons from the planetary motion example
Discussion
- Different perspectives on statistical modeling and prediction
- Justification of iterative model building
- Model selection and overfitting
- Bigger datasets demand bigger models
- Prediction, generalization, and poststratification
- Going forward