## Markov Chain Monte Carlo

 Unit code: MATH48122 Credit Rating: 15 Unit level: Level 4 Teaching period(s): Semester 2 Offered by School of Mathematics Available as a free choice unit?: N

#### Requisites

Prerequisite

MATH48122 pre-requisites

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)

#### 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.

#### 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.

#### Assessment methods

• Other - 50%
• Written exam - 50%

#### Assessment Further Information

• Biweekly courseworks: 50%
• End of semester written examination: two hours 50%

#### 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. Reversible jump MCMC: known number of parameters. [2]
6. Hamiltonian Monte Carlo. [2]
7. Approximate Bayesian Computation: simulation based inference. [4]

• W. R. Gilks, S. Richarson and D. Spiegelhalter, Markov chain Monte Carlo methods in Practice, Chapman and Hall.

#### 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.

#### Study hours

• Lectures - 22 hours
• Tutorials - 22 hours
• Independent study hours - 106 hours

#### Teaching staff

Christiana Charalambous - Unit coordinator