## STAT443 Bayesian Statistics

First Semester |

Bayesian methods provide an approach to statistics that a rapidly-growing number of scientists are starting to use. The key difference between Bayesian and classical statistics is that Bayesian inference makes direct use of probability to represent all uncertainty. In this course we examine the underpinnings of Bayesian inference and familiarize students with computer-based methods used in the Bayesian approach to statistics.

PLEASE NOTE: The contents of this page are meant only as a guideline of what to expect during the paper. The lecturer reserves the right to adjust some details of the paper during the year, as is deemed appropriate.

### Paper details

We begin by comparing and contrasting Bayesian inference with its classical counterpart. We then follow with an introduction to Bayes’ theorem, definitions of probability, prior distributions and posterior inference. Students will work towards developing a Bayesian analysis of their own with real data, which will be submitted as a publication-style report document.

### Potential students

Post-graduate students in statistics.

### Prerequisites

STAT 362. Students should also be familiar with using the R programming language. Those unfamiliar should contact the lecturer to obtain additional support materials.

### Main topics

- Subjective probability, belief, and exchangeability
- Random variables
- Exponential family of distributions and conjugacy
- Monte Carlo approximation
- Inference under the normal model
- Gibbs sampling
- Hierarchical modeling
- Bayesian regression
- Metropolis-Hastings sampling
- Linear mixed effects models
- Latent variable methods
- Model checking

### Suggested texts

Gelman, A., Carlin, J., Stern, H., and Rubin, D.B. (2003) Bayesian Data Analysis. Second Edition

Robert, C. P. (2007) The Bayesian Choice. Second edition

Gelman, A. and Hill, J. (2006) Data Analysis Using Regression and Multilevel/Hierarchical Models

Albert, J. (2009) Bayesian Computation with R

Link, W. A. and Barker, R. J. (2010) Bayesian Inference with Ecological Applications

### Computing Resources

**Editors**

TeXnicCenter (Free LaTeX editor for Windows)

TeXShop (Free LaTeX editor for Macintosh

TextMate (Code editor for Macintosh)

TextWrangler (Free code editor for Macintosh)

GNU Emacs (Free code editor for all platforms)

**LaTeX Distributions**

**R**

The Comprehensive R Archive Network

### Lecturer

Matthew Schofield, Room 237

Peter Dillingham

### Lectures

Two hours of lecture per week, in Room B21 (Science III). Time and location to be arranged.

### Tutorial

One per week at a time to be arranged

### Internal Assessment

The internal assessment is made up of 6 bi-weekly homework assignments.

All assignments and projects must be submitted in electronic format, preferably using the ;http://en.wikipedia.org/wiki/LaTeX">LaTeX document preparation language;. See the course resource page for downloadable documents for learning LaTeX.

### Exam format

Two-hour exam covering all aspects of the course.

### Final mark

Your final mark F in the paper will be calculated according to this formula:

**F = max(E, (2E + A)/3)**

where:

- E is the Exam mark
- A is the Assignments mark

and all quantities are expressed as percentages.

### Students must abide by the University’s Academic Integrity Policy

**Academic integrity** means being honest in your studying and assessments. It is the basis for ethical decision-making and behaviour in an academic context. Academic integrity is informed by the values of honesty, trust, responsibility, fairness, respect and courage.

**Academic misconduct** is seeking to gain for yourself, or assisting another person to gain, an academic advantage by deception or other unfair means. The most common form of academic misconduct is plagiarism.

Academic misconduct in relation to work submitted for assessment (including all course work, tests and examinations) is taken very seriously at the University of Otago.

All students have a responsibility to understand the requirements that apply to particular assessments and also to be aware of acceptable academic practice regarding the use of material prepared by others. Therefore it is important to be familiar with the rules surrounding academic misconduct at the University of Otago; they may be different from the rules in your previous place of study.

Any student involved in academic misconduct, whether intentional or arising through failure to take reasonable care, will be subject to the University’s Student Academic Misconduct Procedures which contain a range of penalties.

If you are ever in doubt concerning what may be acceptable academic practice in relation to assessment, you should clarify the situation with your lecturer before submitting the work or taking the test or examination involved.

Types of academic misconduct are as follows:

**Plagiarism**

The University makes a distinction between unintentional plagiarism (Level One) and intentional plagiarism (Level Two).

- Although not intended,
*unintentional plagiarism*is covered by the Student Academic Misconduct Procedures. It is usually due to lack of care, naivety, and/or to a lack to understanding of acceptable academic behaviour. This kind of plagiarism can be easily avoided. *Intentional plagiarism*is gaining academic advantage by copying or paraphrasing someone elses work and presenting it as your own, or helping someone else copy your work and present it as their own. It also includes self-plagiarism which is when you use your own work in a different paper or programme without indicating the source. Intentional plagiarism is treated very seriously by the University.

**Unauthorised Collaboration**

Unauthorised Collaboration occurs when you work with, or share work with, others on an assessment which is designed as a task for individuals and in which individual answers are required. This form does not include assessment tasks where students are required or permitted to present their results as collaborative work. Nor does it preclude collaborative effort in research or study for assignments, tests or examinations; but unless it is explicitly stated otherwise, each students answers should be in their own words. If you are not sure if collaboration is allowed, check with your lecturer..

**Impersonation**

Impersonation is getting someone else to participate in any assessment on your behalf, including having someone else sit any test or examination on your behalf.

**Falsiﬁcation**

Falsiﬁcation is to falsify the results of your research; presenting as true or accurate material that you know to be false or inaccurate.

**Use of Unauthorised Materials**

Unless expressly permitted, notes, books, calculators, computers or any other material and equipment are not permitted into a test or examination. Make sure you read the examination rules carefully. If you are still not sure what you are allowed to take in, check with your lecturer.

**Assisting Others to Commit Academic Misconduct**

This includes impersonating another student in a test or examination; writing an assignment for another student; giving answers to another student in a test or examination by any direct or indirect means; and allowing another student to copy answers in a test, examination or any other assessment.

**Thomas Bayes**, 1702-1761, who followed his father into the Ministry in England. He was intensely interested in mathematics at a time when calculus was in its infancy. His theory of probability was published in

*Essay towards solving a problem in the doctrine of chances*by the Royal Society of London in 1764 — three years after his death, being submitted by a friend who recognised its value.

His conclusions were accepted by Laplace in 1781, rediscovered by Condorcet, and remained unchallenged until Boole questioned them. Since then Bayes’ techniques have been subject to controversy.

### Some quotable quotes

“...everyone is, should be, or will soon be a Bayesian” (H. Chernoff)“Every statistician would be a Bayesian if he took the trouble to read the literature thoroughly...” (D. V. Lindley)

“Probability does not exist” (B. De Finetti (1974) in

*Theory of Probability*)