MAS371 Applied Probability

Note: Information for future academic years is provisional. Timetable information and teaching staff are especially likely to change, but other details may also be altered, some courses may not run at all, and other courses may be added.

 Semester 2, 2022/23 10 Credits Lecturer: Dr Jonathan Jordan Home page Timetable Reading List Aims Outcomes Teaching Methods Assessment Full Syllabus

This unit will link probability modelling to statistics. It will explore a range of models that can be constructed for random phenomena that vary in time or space - the evolution of an animal population, for example, or the number of cancer cases in different regions of the country. It will illustrate how models are built and how they might be applied: how likelihood functions for a model may be derived and used to fit the model to data, and how the result may be used to assess model adequacy. Models examined will build on those studied in MAS275.

Prerequisites: MAS223 (Statistical Inference and Modelling); MAS275 (Probability Modelling)
No other modules have this module as a prerequisite.

Outline syllabus

• Basic techniques: likelihood functions and their properties and use.
• Continuous time Markov chains: Introduction; generator matrices; informal coverage of stationary distributions and convergence.
• Inference for stochastic processes: deriving likelihood functions for stochastic processes; fitting models to data; model criticism.
• Applications of Markov chains: birth-death processes; queues.
• Point processes: homogeneous and inhomogeneous Poisson processes, spatial and marked point processes, inference for point processes.

Aims

• Illustrate the construction of probability models for random phenomena;
• Introduce some of the common classes of models for random phenomena;
• Illustrate how probability models may be fitted to data;
• Discuss applications of fitted models.

Learning outcomes

• Apply some of the common classes of probability models;
• Formulate and interpret models for random phenomena;
• Use likelihood-based methods to estimate parameters of probability models from data;
• Use statistical methods to investigate how well a model fits to a given data-set.

Teaching methods

Lectures, problem solving

20 lectures, no tutorials

Assessment

One 2 hour written examination. In 2021/22 this will be open book.

Full syllabus

1 Introduction
Types of models, examples of applications.

2 Basic Techniques
Likelihood functions and their properties; information; statement of key theorems.
3 Modelling with Discrete Time Markov chains
Model adequacy; time dependence; estimation of transition probabilities; long-term behaviour.
4 Inference for Discrete Time Markov Chain Models
Properties of MLEs; confidence intervals; testing particular values; goodness of fit; higher order chains.
5 Continuous Time Markov Chain Models
Introduction; generator matrices; calculation of transition probabilities; informal coverage of stationary distributions and convergence; likelihood and inference.
6 Point processes
Homogeneous and inhomogeneous Poisson processes, spatial and marked point processes, inference for point processes.