Preface
About the module
The notes are for the version of MATH27720 Statistics 2 taught in spring 2025 at the University of Manchester.
The MATH27720 Statistics 2 module is designed to run over the course of 10 weeks. It has the following two part structure:
- Entropy and likelihood (W1–W5)
- Bayesian statistics (W6–W10)
This module focuses on conceptual understanding and methods, not on theory. Specifically, you will learn about the foundations of statistical learning using likelihood and Bayesian approaches and also how these are underpinned by entropy.
The presentation in this course is non-technical and most sections and examples will be easily accessible for a year 2 mathematics student. Sections and examples marked \(\color{Red} \blacktriangleright\) are either conceptually or technically a bit more advanced (e.g. involving more complicated matrix operations). These examples may be omitted on first reading.
If you are a University of Manchester student and enrolled in this module you will find additional support material on Blackboard:
- a weekly learning plan,
- worksheets with examples and solutions (and R code), and
- exam papers of previous years.
Furthermore, a MATH27720 Statistics 2 online reading list is hosted by the University of Manchester library.
In a nutshell, the two key aims of the MATH27720 Statistics 2 module are
- to provide a principled introduction to maximum likelihood and Bayesian statistical analysis and
- to demonstrate that statistics offers a well founded and coherent theory of information, rather than just seemingly unrelated collections of “recipes” for data analysis.
The first part of the module (Weeks 1–5) we will explore the method of maximum likelihood both practically and more theoretically in terms of its foundations.
The second part of this module (Weeks 6–10) focuses on the Bayesian approach to statistical estimation and inference that can be viewed as a natural extension of likelihood-based statistical analysis that overcomes some of the limitations of maximum likelihood.
Acknowledgements
These notes are based in part on my earlier notes for MATH20802 Statistical Methods which was last run in Spring 2023. Many thanks to Beatriz Costa Gomes for her help in creating the 2019 version of the lecture notes when I was teaching the MATH20802 module for the first time and to Kristijonas Raudys for his extensive feedback on the 2020 version.