Module Directory

This module focuses principally on Bayesian computational statistics and covers different types of Monte Carlo methods, including rejection sampling, importance sampling and Markov chain Monte Carlo.

This module will also cover generalised linear models (GLM) and built upon GLM, modern machine learning techniques are also introduced in the end.

The aims of this module are:

• To introduce the philosophy of Bayesian statistics;


• To familiarize students with a Monte Carlo approach to Bayesian statistical analysis;


• To develop students' R programming skills;


• To extend understanding of statistical inference, statistical modelling and statistical application;


• To understand the basics of generalised linear regression;


• To understand the basics of machine learning.

By the end of this module, students will be expected to be able to:

1. Have a conceptual understanding of Bayes' theorem and Bayesian statistical modelling.


2. Have a conceptual understanding of Markov chain Monte Carlo simulation and the convergence diagnostic for MCMC.


3. Have a conceptual understanding of rejection sampling and importance sampling.


4. Have ability to develop a Monte Carlo simulation algorithm for simple probability distributions and implement in R.


5. Have a conceptual understanding of a generalised linear model and implement in R.


6. Have a conceptual understanding of elementary principles of Machine Learning and implement in R.

Indicative Syllabus

Monte Carlo and Bayesian computational statistical methods [CS1 5, CM2 5.3]

1. Bayesian statistical methods:likelihood function, prior distribution, posterior distribution.


2. Random variable generation and Monte Carlo integration,Classical Monte Carlo Integration, transformation methods, importance sampling, rejection sampling.


3. Simulation from posterior distribution via Markov chain Monte Carlo:
   Markov chains, stationary distribution, transition probability, general balance, detail balance and the MCMC principle.


4. Metropolis-Hastings algorithm,Independent Metropolis-Hastings algorithm, Random walks Metropolis-Hastings algorithm.


5. Gibbs sampler


6. Diagnostic of MCMC convergence

Generalised linear model [CS1-4.2]

1. Understand fundamental concepts of (GLM)


2. Define an exponential family of distributions. Show that the following distributions may be written in this form: binomial, Poisson, exponential, gamma, normal.


3. State the mean and variance for an exponential family and define the variance function and the scale parameter. Explain what is meant by the link function.


4. Explain what is meant by a variable, a factor taking categorical values. Define the linear predictor, illustrating its form for simple models, including polynomial models and models involving factors.


5. Fit a generalised linear model to a data set in R and interpret the output.

Machine learning [CS2-5]

1. Explain the main branches of machine learning and describe examples of the types of problems typically addressed by Machine Learning.


2. Describe and give examples of key supervised and unsupervised Machine Learning techniques, explaining the difference between regression and classification and between generative and discriminative models.


3. Explain in detail and use appropriate package to apply Machine Learning techniques in R to simple problems.

Teaching in the School will be delivered using a range of face to face lectures, classes and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.