# Module Information

#### Course Delivery

Delivery Type | Delivery length / details |
---|---|

Lecture | 18 Hours. (18 x 1 hour lectures) |

Seminars / Tutorials | 4 Hours. (4 x 1 hour example classes) |

#### Assessment

Assessment Type | Assessment length / details | Proportion |
---|---|---|

Semester Exam | 2 Hours (written examination) | 100% |

Supplementary Assessment | 2 Hours (written examination) | 100% |

### Learning Outcomes

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

1. construct and interpret Classical confidence intervals and tests of hypotheses for a population mean (Normal) and a probability parameter;

2. set up a Bayesian analysis of the same situations;

3. interpret prior and posterior distributions for parameters and construct Bayesian confidence intervals;

4. explain the differences between Classical and Bayesian analyses;

5. extend the ideas to other distributional families.

### Brief description

This module re-examines the ideas of likelihood, confidence intervals and hypothesis testing in Classical Inference and considers their interpretation more deeply. An alternative approach known as Bayesian Inference is introduced in which prior information is modelled in the form of a distribution and updated in the presence of data using Bayes's Theorem. The concepts prior, posterior, predictive and preposterior are introduced. Applications to inferences about a (Normal) population mean, a (Binomial) probability parameter and other distributional families are discussed in detail. The meanings and interpretations of the two approaches are discussed at length.

### Aims

To introduce the basic ideas and concepts of statistical inference.

### Content

2. BAYESIAN INFERENCE Bayes' Theorem. Prior and posterior odds. Prior and posterior distributions. Conjugate families. Prior knowledge and prior ignorance. Quantification of knowledge. Predictive distributions. Preposterior distributions. Bayesian point estimation, loss functions.

3. CONFIDENCE STATEMENTS Classical: pivotal functions, confidence intervals. Bayesian: highest density intervals, predictive intervals. Interpretation of relative likelihood intervals.

4. HYPOTHESIS TESTING Classical: null and alternative hypotheses. Neyman Pearson theory. UMP tests.

5. OVERVIEW Comparisons between Classical and Bayesian approaches.

### Reading List

**Recommended Text**

Hogg, Robert V. (c2006.) Probability and statistical inference /Robert V. Hogg, Elliot A. Tanis. Prentice Hall Primo search

**Supplementary Text**

Freund, John E. (c2004.) John E. Freund's mathematical statistics with applications /John E. Freund. Prentice Hall Primo search Garthwaite, Paul H. (c2002 (2006 pri) Statistical inference /Paul Garthwaite, Ian Jolliffe, and Byron Jones. Oxford University Press Primo search Gelman, Carlin, Stern, Rubin (c2004.) Bayesian data analysis /Andrew Gelman ... [et al.]. Chapman & Hall/CRC Primo search Larsen, Richard J. (2005.) An introduction to mathematical statistics /Richard J. Larsen, Morris L. Marx. Prentice Hall Primo search Lee, Peter M. (1997 (2001 prin) Bayesian statistics :an introduction /Peter M. Lee. Arnold Primo search Lindgren, B. W. (c1993.) Statistical theory /Bernard W. Lindgren. Chapman & Hall Primo search Wackerly, Dennis (Sept. 2007) Mathematical Statistics W/Applications Brooks/Cole Primo search

**Recommended Background**

V D Barnett Comparative Statistical Inference Wiley Primo search

### Notes

This module is at CQFW Level 6