Bayesian methods are now increasingly widely in data analysis across most scientific research fields. Given that Bayesian methods differ conceptually and theoretically from their classical statistical counterparts that are traditionally taught in statistics courses, many researchers do not have opportunities to learn the fundamentals of Bayesian methods, which makes using Bayesian data analysis in practice more challenging. The aim of this course is to provide a solid introduction to Bayesian methods, both theoretically and practically. We will begin by teaching the fundamental concepts of Bayesian inference and Bayesian modelling, including how Bayesian methods differ from their classical statistics counterparts, and show how to do Bayesian data analysis in practice in R. We then provide a solid introduction to Bayesian approaches to these topics using R and the brms package. We begin by covering Bayesian approaches to linear regression. We will then proceed to Bayesian approaches to generalized linear models, including binary logistic regression, ordinal logistic regression, Poisson regression, zero-inflated models, etc. Finally, we will cover Bayesian approaches to multilevel and mixed effects models. Throughout this course, we will be using, via the brms package, Stan based Markov Chain Monte Carlo (MCMC) methods.

Intended Audience

This course is aimed at anyone who is interested to learn and apply Bayesian data analysis in any area of science, including the social sciences, life sciences, physical sciences. No prior experience or familiarity with Bayesian statistics is required.

Teaching Format

This course will be practical, hands-on, and workshop based. For some topics, there will a very minimal amount of lecture style presentations, i.e., using slides or blackboard, to introduce and explain key concepts and theories, but almost all of our time will be spent doing data wrangling using R. Any code that the instructor produces during these sessions will be uploaded to a publicly available GitHub site after each session.

The course will take place online using Zoom. On each day, the live video broadcasts will occur between 5pm-10pm (GMT, UTC).

All sessions will be video recorded and made available to all attendees as soon as possible.

Although not strictly required, using a large monitor or preferably even a second monitor will make the learning experience better, as you will be able to see my RStudio and your own RStudio simultaneously.

Assumed quantitative knowledge

We assume familiarity with inferential statistics concepts like hypothesis testing and statistical significance, and some practical experience with commonly used methods like linear regression, correlation, or t-tests. Most or all of these concepts and methods are covered in a typical undergraduate statistics courses in any of the sciences and related fields.

Assumed computer background

R experience is desirable but not essential. Although we will be using R extensively, all the code that we use will be made available, and so attendees will just need to copy and paste and add minor modifications to this code. Attendees should install R and RStudio and some R packages on their own computers before the workshops, and have some minimal familiarity with the R environment.

Equipment and software requirements

A computer with a working version of R or RStudio is required. R and RStudio are both available as free and open source software for PCs, Macs, and Linux computers. In addition to R and RStudio, some R packages, particularly brms, are required. Instructions on how to install the necessary software will be provided to the attendees before the course begins.

Course programme

Day 1

• Topic 1: We will begin with a overview of what Bayesian data analysis is in essence and how it fits into statistics as it practiced generally. Our main point here will be that Bayesian data analysis is effectively an alternative school of statistics to the traditional approach, which is referred to variously as the classical, or sampling theory based, or frequentist based approach, rather than being a specialized or advanced statistics topic. However, there is no real necessity to see these two general approaches as being mutually exclusive and in direct competition, and a pragmatic blend of both approaches is entirely possible.

• Topic 2: Introducing Bayes’ rule. Bayes’ rule can be described as a means to calculate the probability of causes from some known effects. As such, it can be used as a means for performing statistical inference. In this section of the course, we will work through some simple and intuitive calculations using Bayes’ rule. Ultimately, all of Bayesian data analysis is based on an application of these methods to more complex statistical models, and so understanding these simple cases of the application of Bayes’ rule can help provide a foundation for the more complex cases.

• Topic 3: Bayesian inference in a simple statistical model. In this section, we will work through a classic statistical inference problem, namely inferring the number of red marbles in an urn of red and black marbles, or equivalent problems. This problem is easy to analyse completely with just the use of R, but yet allows us to delve into all the key concepts of all Bayesian statistics including the likelihood function, prior distributions, posterior distributions, maximum a posteriori estimation, high posterior density intervals, posterior predictive intervals, marginal likelihoods, Bayes factors, model evaluation of out-of-sample generalization.

Day 2

• Topic 4: Bayesian analysis of normal models. Statistical models based on linear and normal distribution are a mainstay of statistical analyses in general. They encompass models such as linear regression, Pearson’s correlation, t-tests, ANOVA, ANCOVA, and so on. In this section, we will describe how to do Bayesian analysis of normal linear models, focusing on simple examples. One of the aims of this section is to identify some important and interesting parallels between Bayesian and classical or frequentist analyses. This shows how Bayesian and classical analyses can be seen as ultimately providing two different perspectives on the same problem.

• Topic 5: The previous section provides a so-called analytical approach to linear and normal models. This is where we can calculate desired quantities and distributions by way of simple formulae. However, analytical approaches to Bayesian analyses are only possible in a relatively restricted set of cases. On the other hand, numerical methods, specifically Markov Chain Monte Carlo (MCMC) methods can be applied to virtually any Bayesian model. In this section, we will re-perform the analysis presented in the previous section but using MCMC methods. For this, we will use the brms package in R that provides an exceptionally easy to use interface to Stan.

Day 3

• Topic 6: Bayesian linear models. We begin by covering Bayesian linear regression. For this, we will use the brm command from the brms package, and we will compare and contrast the results with the standard lm command. By comparing and contrasting brm with lm we will see all the major similarities and differences between the Bayesian and classical approach to linear regression. We will, for example, see how Bayesian inference and model comparison works in practice and how it differs conceptually and practically from inference and model comparison in classical regression. As part of this coverage of linear models, we will also use categorical predictor variables and explore varying intercept and varying slope linear models.

Day 4

• Topic 7: Extending Bayesian linear models. Classical normal linear models are based on strong assumptions that do not always hold in practice. For example, they assume a normal distribution of the residuals, and assume homogeneity of variance of this distribution across all values of the predictors. In Bayesian models, these assumptions are easily relaxed. For example, we will see how we can easily replace the normal distribution of the residuals with a t-distribution, which will allow for a regression model that is robust to outliers. Likewise, we can model the variance of the residuals as being dependent on values of predictor variables.

• Topic 8: Bayesian generalized linear models. Generalized linear models include models such as logistic regression, including multinomial and ordinal logistic regression, Poisson regression, negative binomial regression, zero-inflated models, and other models. Again, for these analyses we will use the brms package and explore this wide range of models using real world data-sets. In our coverage of this topic, we will see how powerful Bayesian methods are, allowing us to easily extend our models in different ways in order to handle a variety of problems and to use assumptions that are most appropriate for the data being modelled.

Day 5

• Topic 9: Multilevel and mixed models. In this section, we will cover the multilevel and mixed effects variants of the regression models, i.e. linear, logistic, Poisson etc, that we have covered so far. In general, multilevel and mixed effects models arise whenever data are correlated due to membership of a group (or group of groups, and so on). For this, we use a wide range of real-world data-sets and problems, and move between linear, logistic, etc., models are we explore these analyses. We will pay particular attention to considering when and how to use varying slope and varying intercept models, and how to choose between maximal and minimal models. We will also see how Bayesian approaches to multilevel and mixed effects models can overcome some of the technical problems (e.g. lack of model convergence) that beset classical approaches.

GitHub resources

Further resources for this training course can be found on Github at mark-andrews/bada01.