General description
This module re-examines the ideas of confidence intervals and hypothesis testing in Classical Inference and considers their interpretation more deeply. An alternative approach known as Bayesian Inference is introduced and developed, and consideration given to the formal description of prior information along with the way this information is modified in the presence of data. The concepts prior, posterior, predictive and preposterior are introduced. Applications to inferences about a (Normal) population mean and a (Binomial) probability parameter are discussed in detail, and extensions to other distributional families indicated. [The meanings and interpretations of the two approaches are discussed at length, along with the philosophical bases of other forms of statistical inference, such as the fiducial and likelihood approaches.]

Aims
To introduce the statistical ideas involved in designing efficient experiements and in the analysis of results.

Learning outcomes
On completion of this module, a student should be able to:

• construct and interpret Classical confidence intervals for a population mean (Normal) and a probability parameter;
• set up a Bayesian analysis of the same situation;
• interpret prior and posterior distributions for parameters and construct Bayesian confidence intervals;
• explain the differences between Classical and Bayesian analyses;
• extend the ideas to other distributional families;
• explain the bases of other forms of inference such as the fiducial and likelihood approaches.

Syllabus
1. CLASSICAL INFERENCE: Basic aims and concepts. Point estimation. Interval estimation and hypothesis testing. Uniformly most powerful tests.
2. BAYESIAN INFERENCE: Thomas Bayes. Prior and posterior distributions. Odds ratios. Prior ignorance and prior knowledge. Quantification of knowledge. Bayesian confidence intervals. Predictive distributions. Predictive intervals. Preposterior distributions.
3. OTHER APPROACHES: Decision Theory. Fiducial Inference. Likelihood Inference. Information.