# APTS module: Statistical Inference

** Module leader**: J C Rougier

*Please see the full Module Specifications for background information relating to all of the APTS modules, including how to interpret the information below.*

**Aims**: To explore the basic principles of statistical inference: its origins in decision support, the nature of statistical parameters, the different viewpoints of Bayesian and Frequentist approaches, and the meaning and value of ubiquitous constructs such as *p*-values, confidence sets, and hypothesis tests.

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

**Preliminaries**: Students should have done at least one course in probability and one in statistics. Preliminary reading will cover the necessary material on probability. For statistics, students should be familiar with: the idea of a statistical model, statistical parameters, the likelihood function, estimators, the maximum likelihood estimator, confidence intervals and hypothesis tests, *p*-values, Bayesian inference, prior and posterior distributions.

Further information on all of these topics can be found in standard undergraduate statistics textbooks, for example

- J.A. Rice, 1999, Mathematical Statistics and Data Analysis, 2nd edn, Duxbury Press (more recent edition available); or
- Morris H, DeGroot, and Mark J Schervish, 2002, Probability & Statistics, Addison Wesley, 3rd edn. Prof. Schervish maintains a list of errata at http://www.stat.cmu.edu/~mark/degroot/index.html.

More advanced treatments can be found in

- G.A. Young and R.L. Smith, 2005, Essential of Statistical Inference, Cambridge University Press.
- A.C. Davison, 2003, Statistical Models, Cambridge University Press. This book also contains a wealth of applications. Prof. Davison maintains a list of errata at http://statwww.epfl.ch/davison/SM/.

**Topics**:

- What is statistics? Statistical models, prediction and inference, Frequentist and Bayesian approaches.
- Principles of inference: the Likelihood Principle, Birnbaum'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.

**Assessment**: Exam-style questions on the implementation of different approaches in particular types of inference, possibly involving additional reading.