# MSc Statistics

Year of entry: 2021

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## Course unit details:Markov Chain Monte Carlo

Unit code MATH68122 15 FHEQ level 7 – master's degree or fourth year of an integrated master's degree Semester 2 Department of Mathematics No

### Overview

Since the late 1980's MCMC has been widely used in statistics and the range of its applications are ever increasing. This course will introduce MCMC methodology, in particular, the Metropolis-Hastings algorithm which is the basis for all MCMC. The implementation of MCMC will be discussed in detail with numerous examples.

### Pre/co-requisites

Unit title Unit code Requirement type Description
Probability 2 MATH20701 Pre-Requisite Compulsory
Statistical Methods MATH20802 Pre-Requisite Compulsory
Statistical Computing MATH48091 Pre-Requisite Recommended

Students are not permitted to take, for credit, MATH48122 in an undergraduate programme and then MATH68122 in a postgraduate programme at the University of Manchester, as the courses are identical.

### Aims

To introduce the student to computational Bayesian statistics, in particular Markov chain Monte Carlo (MCMC)

### Learning outcomes

On successful completion of this course unit students will be able to:

1. Apply various MCMC algorithms, such as the Metropolis-Hastings algorithm and the Gibbs sampler, to obtain samples from complex distributions and for parameter estimation in standard problems, such as regression modelling;

2. employ the Approximate Bayesian Computation (ABC) algorithm for parameter estimation in various problems, including population genetic models;

3. describe the various algorithms in words but also implement the algorithms through statistical software;

4. evaluate the performance of the algorithms using diagnostics and explain how to improve performance when necessary;

5. formulate MCMC/ABC algorithms to perform model selection.

### Syllabus

1. Introduction: Bayesian statistics, Markov chains. [2]
2. Gibbs Sampler: data augmentation, burn-in, convergence. [4]
3. Metropolis-Hastings algorithm: independent sampler, random walk Metropolis, scaling, multi-modality. [4]
4. MCMC Issues: Monte Carlo Error, reparameterisation, hybrid algorithms, convergence diagnostics. [4]
5. Perfect Simulation. [2]
6. Reversible jump MCMC: unknown number of parameters. [2]
7. Approximate Bayesian Computation: simulation based inference. [4]

### Assessment methods

Method Weight
Other 50%
Written exam 50%
• Biweekly courseworks: 50%
• End of semester written examination: 50%

### Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.

• W. R. Gilks, S. Richarson and D. Spiegelhalter, Markov chain Monte Carlo methods in Practice, Chapman and Hall.
• A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Chapman and Hall.

### Study hours

Scheduled activity hours
Lectures 24
Practical classes & workshops 22
Independent study hours
Independent study 104

### Teaching staff

Staff member Role
Matthew Thorpe Unit coordinator