MMath Mathematics and Statistics

The unit aims to introduce students to the fundamentals of Bayesian inference and the computational techniques used to apply it in data analysis and model evaluation. 

MATH48221 Pre-Requisites: MATH27720 and MATH27711

The unit aims to:  

Introduce students to the fundamentals of Bayesian inference and the computational techniques used to apply it in data analysis and model evaluation. 

On the successful completion of the course, students will be able to:

1. Derive posterior distributions for exact Bayesian inference and make inferences/predictions based on these posteriors;
2. Apply various computational algorithms to obtain samples from complex posterior distributions and for parameter estimation;
3. Describe the various algorithms in words but also implement key algorithms through statistical software;
4. Make informed choices among available algorithms for practical data analysis;
5. Solve statistical modelling and inference problems within the Bayesian paradigm.

Part A – Foundation of Bayesian Inference

• Bayesian inference concepts: single and multiple parameter prior and posterior distributions; conjugacy and non-conjugacy; Bayesian estimators; credible intervals.
• Model checking & model comparison: Posterior predictive distribution; Bayesian forecasting; model comparison based on predictive performance; model comparison criteria such as Bayes factors, BIC and DIC; Bayesian Decision Theory; Laplace's approximation.

Part B - Computational Bayesian Statistics

• Gibbs Sampler: data augmentation; burn-in; convergence.
• Metropolis-Hasting’s algorithm: independent sampler; random walk Metropolis; scaling; multi-modality.  
• MCMC Issues: Monte Carlo Error (batch means/window estimates for MCSE); reparameterization; hybrid algorithms; convergence diagnostics for single/multiple chains.  
• Hamiltonian Monte Carlo.
• Approximate Bayesian Inference.

Teaching is composed of two hours of lectures and one example/computer class per week. Teaching materials will be made available online for reference and review. 

Christensen, R., Johnson, W., Branscum, A., & Hanson, T. E. (2010). Bayesian ideas and data analysis: an introduction for scientists and statisticians. CRC press.

 

Heard, N. (2021). An introduction to Bayesian inference, methods and computation. Cham: Springer.

 

McElreath, R. (2020). Statistical rethinking: A Bayesian course with examples in R and Stan, (2nd edn). Chapman and Hall/CRC.

 

Wang, X., Yue, Y. R., & Faraway, J. J. (2018). Bayesian regression modeling with INLA. Chapman and Hall/CRC.