Bayesian Statistics

MATH68221 Bayesian Statistics 15 Semester 1 Postgraduate Taught Level 6 Taban Baghfalaki Unit coordinator School of Mathematics 1 FHEQ level (Framework for Higher Education Qualifications) ' Masters/Integrated Masters P4 ' European Credit Transfer & Accumulation System Rating : 7.5 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.  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.  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:<ul><li>Explain the fundamental concepts of Bayesian inference, including priors, posteriors, conjugacy, non-conjugacy, credible intervals, and Bayesian decision theory.Describe model checking and comparison criteria such as Bayes factors, BIC, DIC, and LPML.</li><li>Implement computational algorithms, including Gibbs sampling, Metropolis-Hastings, Hamiltonian Monte Carlo, and approximate Bayesian methods, to estimate posterior distributions.Fit and interpret Bayesian linear regression, generalized linear models (GLMs), and hierarchical/mixed models using statistical software (R, JAGS, Stan, or equivalent).</li><li>Assess convergence, reliability, and Monte Carlo error in MCMC simulations.Critically evaluate model fit and predictive performance using posterior predictive checks and Bayesian model selection criteria.</li><li>Compare alternative Bayesian models and algorithms for practical data analysis tasks. Justify prior choices, model structures, and algorithm selection for real-world datasets.</li><li>Construct original Bayesian models for novel datasets, including specifying likelihoods, priors, and hierarchical structures.Develop complete Bayesian workflows from model specification, through computation and inference, to reporting results and recommendations.</li></ul> Part A – Foundation of Bayesian Inference<ul><li>Bayesian inference concepts: single and multiple parameter prior and posterior distributions; conjugacy and non-conjugacy; Bayesian estimators; credible intervals.</li><li>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.  </li></ul>Part B - Computational Bayesian Statistics<ul><li>Gibbs Sampler: data augmentation; burn-in; convergence. Metropolis-Hasting’s algorithm: independent sampler; random walk Metropolis; scaling; multi-modality.  </li><li>MCMC Issues: Monte Carlo Error (batch means/window estimates for MCSE); reparameterization; hybrid algorithms; convergence diagnostics for single/multiple chains.  </li><li>Hamiltonian Monte Carlo.</li><li>Approximate Bayesian Inference.</li></ul> 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.  0 Other 25% 1 Written exam 60% 2 Written assignment (inc essay) 15% MATH27720 Probability and Statistics 2 Pre-Requisite Recommended N 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.  Lectures 22 Tutorials 11 0 117