Aims

To introduce and reinforce a range of concepts in probability and statistics with particular emphasis on illustrations in R, including methods that will be useful towards future project work. To reinforce the computing in R studied within MAS1802, and to move towards expectations of more independent programming.

Students will also learn about the Bayesian approach to statistical analysis. Students will be able to explain the distinctive features of Bayesian methodology, understand the role of prior distributions and compute posterior distributions in simple cases.

Module summary

Computational methods are of great use in a wide range of applications of probability and statistics. This module builds on the probability introduced in MAS1604 and the use of R introduced in MAS1802. Students will be introduced to additional concepts and techniques, some of increasing mathematical and computational sophistication. In implementing these methods, students will attain a deeper understanding of foundational probability and statistics, increasing competence with mathematical/statistical computing, and an increasing ability to use such methods independently, towards project-orientated goals.

The second part of the module will be devoted to an introduction to Bayesian methods, in which the prior and posterior distributions of a scalar parameter will be defined. The use of the likelihood to allow the prior distribution to be updated to the posterior distribution will be discussed. The use of Bayes’ theorem to compute posterior distributions from given priors and likelihoods will be described, with particular emphasis given to the case of conjugate distributions.

Outline Of Syllabus

Review of probability ideas: illustrations of properties of univariate, bivariate and trivariate distributions, including use of conditional distributions. Transformations of random variables. Sampling distributions. Illustration of properties of hypothesis tests and confidence intervals.

Introduction to the Bayesian approach: subjective probability; likelihood; sufficiency. Inference for populations using random samples and conjugate priors, including posterior estimates and highest density intervals: inference for the mean of a normal distribution with known variance; inference for parameters in other commonly used distributions. Sequential use of Bayes' Theorem. Parameter constraints. Mixture prior distributions. Asymptotic posterior distribution.

Teaching Rationale And Relationship

Non-synchronous online materials are used for the delivery of theory and explanation of methods, illustrated with examples, and for giving general feedback on assessed work. Present-in-person and synchronous online sessions are used to help develop the students’ abilities at applying the theory to solving problems and to identify and resolve specific queries raised by students, and to allow students to receive individual feedback on marked work. Students who cannot attend a present-in-person session will be provided with an alternative activity allowing them to access the learning outcomes of that session. In addition, office hours/discussion board activity will provide an opportunity for more direct contact between individual students and the lecturer:  a typical student might spend a total of one or two hours over the course of the module, either individually or as part of a group.
Alternatives will be offered to students unable to be present-in-person due to the prevailing C-19 circumstances.
Student’s should consult their individual timetable for up-to-date delivery information.

Assessment Rationale And Relationship

A substantial formal examination is appropriate for the assessment of the material in this module. The course assessments will allow the students to develop their problem solving techniques, to practise the methods learnt in the module, to assess their progress and to receive feedback; these assessments have a secondary formative purpose as well as their primary summative purpose.