This module introduces the expectation-maximisation algorithm, an iterative algorithm for obtaining the maximum likelihood estimate of parameters in problems with intractable likelihoods. Students will explore the use of Markov chain Monte Carlo (MCMC) methods, and will discover the features of the Metro-Hastings algorithm, with emphasis on the Gibbs sampler, independence sampler and random walk Metropolis. Whilst relating to this, students will consider how such methods are closely integrated with Bayesian modelling techniques such as hierarchal modelling, random effects and mixture modelling.

Data augmentation will receive recurring coverage over the course of the module. Students will also gain transferrable knowledge of the usefulness of computers in assisting statistical analysis of complex methods, in addition to experience with the computer statistical package R.

The three month dissertation period (mid-June to mid-September) will involve the application of statistical methodology to a substantive problem. This dissertation is written by the student under the direction of a supervisor. Some projects are collaborative: examples include GlaxoSmithKline; AstraZeneca; Wrightington Hospital; Royal Lancaster Infirmary; Leahurst Veterinary Centre; and the Department of the Environment.

Students will gain a thorough understanding of advanced statistical methods which go beyond the scope of MSc taught components, and will learn about the development of original statistical methodology which will contribute to a fuller understanding of existing methodology. Students are required to make innovative use of the statistical method, leading to substantive findings which would not readily be obtainable by routine application of standard techniques.

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This module is only core for those with the required mathematical background to complete it. Some students may require an introduction to the area, at the graduate level, and they will study the core module titled ‘Statistical Fundamentals I’. If you complete this module, you will not be required to take Statistical Fundamentals I.

The areas that will be covered are statistical inference using maximum likelihood and generalised linear models (GLMs). Building on an undergraduate-level understanding of mathematics, statistics (hypothesis testing and linear regression) and probability (univariate discrete and continuous distributions; expectations, variances and covariances; the multivariate normal distribution), this module will motivate the need for a generic method for model fitting and then demonstrate how maximum likelihood provides a solution to this. Following on from this, GLMs, a widely and routinely used family of statistical models, will be introduced as an extension of the linear regression model.

This module will develop the core topic of maximum likelihood inference previously introduced in MATH501 Statistical Fundamentals I by expanding on numerical and theoretical aspects. Numerical aspects will include obtaining the maximum likelihood estimate using numerical optimisation functions in R, and using the profile likelihood function to obtain both the maximum likelihood estimate and confidence intervals. Theoretical elements covered will include derivation of asymptotic distributions for the maximum likelihood estimator, deviance and profile deviance.

The second half of the module will introduce Bayesian inference as an alternative to maximum likelihood inference. Building on existing knowledge of the likelihood function, the prior and posterior distributions will be introduced. For simple models, analytical forms for the posterior distribution will be introduced and point estimates for the parameter obtained. For more complex models, numerical methods of sampling from the posterior distribution will be demonstrated.

This module provides a practical introduction to statistical learning from model training to deployment. The student will learn about optimisation as a means for model training; the statistical principles and methods that underpin model selection, and the application of classification and regression models to real data problems.

A variety of supervised learning models and their estimation will be covered, including linear models and their connection to kernel based methods; feed-forward neural networks; and tree based models. Tree based models will be extended to their use in random forests and gradient boosted models. In addition topics relevant to big-data problems including dimensionality reduction variable selection; and stochastic gradient descent will be covered.

Students will be introduced to the widely-used statistical computing package R, which will be the primary tool for data analysis and modelling in this module. In addition to learning how to use R effectively and efficiently, students will also be encouraged to compare and contrast with their existing or developing knowledge of general-purpose languages such as Python.

The aim of this module is to provide students with a range of skills that are necessary for applied statistical work including team-working, oral presentation, statistical computing, and the preparation of written reports including statistical analyses. All students will obtain a thorough grasp of R (including R objects and functions, graphs, basic simulations and programming) and be given an introduction to a second statistical computing package.

Students will also learn how to utilise LaTex for writing a complex and structured scientific report that may include mathematical formulae, tables and figures, as well as learn the intricacies of effective scientific writing style such as grammar, referencing, and the presentation of results in appropriate tables and graphs. They will enhance their oral presentation technique using LaTex Beamer to create slides that include complex mathematical formulae, as well as embark on an in-depth team project using Git, R Markdown or iPython notebooks.