# APTS Introductory notes on Statistical Inference

*November 2021*

# Chapter 1 Introduction to the course

These introductory notes are designed to help students prepare for the APTS module on Statistical Inference. The aim is to introduce the idea of a statistical model, to motivate some principles of statistical inference and to review two approaches to statistical inference: the classical approach, also called the frequentist approach, and the Bayesian approach. A recommended book which covers both introductory ideas to statistical inference and elements of this course is Casella and Berger (2002).

**Course aims:** To explore a number of statistical principles, such as the likelihood principle and sufficiency principle, and their logical implications for statistical inference. To consider the nature of statistical parameters, the different viewpoints of Bayesian and Frequentist approaches and their relationship with the given statistical principles. To introduce the idea of inference as a statistical decision problem. To understand the meaning and value of ubiquitous constructs such as p-values, confidence sets, and hypothesis tests.

**Course learning outcomes:** An appreciation for the complexity of statistical inference, recognition of its inherent subjectivity and the role of expert judgement, the ability to critique familiar inference methods, knowledge of the key choices that must be made, and scepticism about apparently simple answers to difficult questions.

The course will cover three main topics:

- Principles of inference: the Likelihood Principle, Birnbaumâ€™s Theorem, the Stopping Rule Principle, implications for different approaches.
- Decision theory: Bayes Rules, admissibility, and the Complete Class Theorems. Implications for point and set estimation, and for hypothesis testing.
- Confidence sets, hypothesis testing, and p-values. Good and not-so-good choices. Level error, and adjusting for it. Interpretation of small and large p-values.

These notes could not have been prepared without, and have been developed from, those prepared by Jonathan Rougier (University of Bristol) who lectured this course previously. I thus acknowledge his help and guidance though any errors are my own.