This course introduces students to the Bayesian paradigm and different approaches to posterior inference.

The course mostly consists of asynchronous content.

Bayesian Statistics

Bayesian Statistics (Level M)

Advanced Bayesian Methods

Advanced Bayesian Methods (Level M)

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100% Continuous Assessment

This will typically be made up of a project, assessed in terms of code and a report (40%) and three homework exercises, including online quizzes  (60%). Full details are provided in the programme handbook.

The aims of this course are:

■ to develop the foundations of modern Bayesian statistics;

■ to demonstrate how prior distributions are updated to posterior distributions in simple statistical models;

■ to introduce students to advanced stochastic simulation methods such as Markov-Chain Monte Carlo;

■ to illustrate how to fit Bayesian models using high-level software for Bayesian hierarchical modelling such as BUGS or STAN as well as implementing stochastic simulation methods such as Markov Chain Monte Carlo (MCMC).

■ to set out how different models can be compared in a Bayesian setting.

By the end of this course students will be able to:

■ Derive, describe and apply the rules for updating prior distributions in the presence of data, and for calculating posterior predictive distributions;

■ Describe and compute various summaries of the posterior distribution, including posterior mean, MAP estimate, posterior standard deviation and credible regions (including HPDRs) and the predictive distribution;

■ Explain the role of hyperparameters in Bayesian inference, introduce them appropriately into statistical models and use the empirical Bayes approach for their determination;

■ Explain the operation and basic theory of the two main Markov-Chain Monte-Carlo methods, Gibbs sampling and the Metropolis-Hastings algorithm;

■ Derive the full conditional distributions for parameters in simple low-dimensional problems;

■ Perform inference in Bayesian models using high-level software such BUGS or STAN as well as implementing Gibbs sampling and the Metropolis-Hastings algorithm in R;

■ Apply diagnostic procedures to check convergence and mixing of MCMC methods

■ Implement Bayesian approaches to model selection.

■ Describe alternative approaches to Bayesian computation such as variational inference and explain their use in simple examples.

Students must submit at least 75% by weight of the components (including examinations) of the course's summative assessment.