## MA322-7-SP-CO:

Bayesian and Computational Statistics

## Key module for

MSC G30012 Statistics,

DIP N32309 Actuarial Science,

MSC N32312 Actuarial Science,

MSC N32324 Actuarial Science,

MPHDG30048 Statistics,

PHD G30048 Statistics,

MPHDN32348 Actuarial Science,

PHD N32348 Actuarial Science

## Module description

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.

## Module aims

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.
- To understand the basic of model selection and regulation.

## Module learning outcomes

By the end of this modules, students will be expeceted to:

- Have a comprehensive understanding of Bayes' theorem and Bayesian statistical modelling.
- Have a comprehensive understanding of Markov chain Monte Carlo (MCMC) simulation and the convergence diagnostic for MCMC.
- Have a comprehensive understanding of rejection sampling and importance sampling.
- Have a critical awareness and ability to develop a Monte Carlo simulation algorithm for simple probability distributions and implement in R.
- Have a comprehensive understanding of a generalised linear model (GLM) and implement in R.
- Have a comprehensive understanding of elementary principles of Machine Learning and implement in R.
- Have a comprehensive understanding of model selection and regulation under Machine Learning framework and implement in R.

## Module information

Syllabus

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

- Bayesian statistical methods:likelihood function, prior distribution, posterior distribution.
- Random variable generation and Monte Carlo integration,Classical Monte Carlo Integration, transformation methods, importance sampling, rejection sampling.
- Simulation from posterior distribution via Markov chain Monte Carlo:

Markov chains, stationary distribution, transition probability, general balance, detail balance and the MCMC principle. - Metropolis-Hastings algorithm,Independent Metropolis-Hastings algorithm, Random walks Metropolis-Hastings algorithm.
- Gibbs sampler
- Diagnostic of MCMC convergence

**Generalised linear model [CS1-4.2]**

- Understand fundamental concepts of (GLM)
- Define an exponential family of distributions. Show that the following distributions may be written in this form: binomial, Poisson, exponential, gamma, normal.
- 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.
- 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.
- Fit a generalised linear model to a data set in R and interpret the output.

** ****Machine learning [CS2-5]**

- Explain the main branches of machine learning and describe examples of the types of problems typically addressed by Machine Learning.
- 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.
- Explain in detail and use appropriate package to apply Machine Learning techniques in R to simple problems.

## Learning and teaching methods

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.

## Bibliography*

## Assessment items, weightings and deadlines

Coursework / exam | Description | Deadline | Coursework weighting |
---|---|---|---|

Coursework | Test | ||

Exam | Main exam: In-Person, Open Book (Restricted), 180 minutes during Summer (Main Period) | ||

Exam | Reassessment Main exam: In-Person, Open Book (Restricted), 180 minutes during September (Reassessment Period) |

### Exam format definitions

**Remote, open book:**Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.**In-person, open book:**Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.**In-person, open book (restricted):**The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.**In-person, closed book:**The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary, for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.

Your department will provide further guidance before your exams.

### Overall assessment

Coursework | Exam |
---|---|

30% | 70% |

### Reassessment

Coursework | Exam |
---|---|

30% | 70% |

## External examiner

**1289**hours,

**36 (2.8%)**hours available to students:

**1253**hours not recorded due to service coverage or fault;

**0**hours not recorded due to opt-out by lecturer(s).

*** Please note:** due to differing publication schedules, items marked with an asterisk (*) base their information upon the previous academic year.

**Disclaimer:** The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can
be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements,
industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist
of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules.
The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.

The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.