Bayesian Data Analysis (BADA01) This course will be delivered live
10th - 14th January 2022
https://www.prstatistics.com/course/bayesian-data-analysis-bada01/
Classes from 12:00-17:00 Eastern Standard Time
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Bayesian methods are now increasingly used 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.
Day 1 ? 12:00-17:00
? 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 12:00-17:00
? 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 12:00-17:00
? 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 12:00-17:00
? 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 12:00-17:00
? 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.
Oliver Hooker PhD.
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