# 732A91 Bayesian Learning

### Course information

#### Aims

The course aims to give a solid introduction to the Bayesian approach to statistical inference, with a view towards applications in data mining and machine learning. After an introduction to the subjective probability concept that underlies Bayesian inference, the course moves on to the mathematics of the prior-to-posterior updating in basic statistical models, such as the Bernoulli, normal and multinomial models. Linear regression and spline regression are also analyzed using a Bayesian approach. The course subsequently shows how complex models can be analyzed with simulation methods like Markov Chain Monte Carlo (MCMC). Bayesian prediction and marginalization of nuisance parameters is explained, and introductions to Bayesian model selection and Bayesian decision theory are also given.

#### Contents

- Introduction to subjective probability and the basic ideas behind Bayesian inference
- Prior-to-posterior updating in basic statistical models, such as the Bernoulli, normal and multinomial models.
- Bayesian analysis of linear and nonlinear regression models
- Shrinkage, variable selection and other regularization priors
- Bayesian analysis of more complex models with simulation methods, e.g. Markov Chain Monte Carlo (MCMC).
- Bayesian prediction and marginalization of nuisance parameters
- Introduction to Bayesian model selection
- Introduction to Bayesian decision theory.

#### Intended audience and admission requirements

This course is given primarily for students on the Master's programme

*Statistics and Machine Learning*. It is also offered to Master students in other subjects and to interested Ph.D. students (with a more advanced examination).

Students admitted to the Master's programme in Statistics and Machine Learning fulfill the admission requirements for the course.

Students not admitted to the Master's programme in Statistics and Machine Learning should have passed:

- an intermediate course in probability and statistical inference
- a basic course in mathematical analysis
- a basic course in linear algebra
- a basic course in programming

#### Course plan

The TimeEdit schedule for the course is available here.

The labs should be done in pairs of students whereas the project is individual work.

All labs and the project should be submitted as PDFs through LISAM.

#### Module 1 - The Bayesics

**Lecture 1**: Basics concepts. Likelihood. The Bernoulli model. The Gaussian model.

**Read**: BDA Ch. 1, 2.1-2.5 | Slides

**Code**: Beta density | Bernoulli model | One-parameter Gaussian model

**Lecture 2**: Conjugate priors. The Poisson model. Prior elicitation. Noninformative priors.

**Read**: BDA Ch. 2.6-2.9 | Slides

**Lecture 3**: Multi-parameter models. Marginalization. Multinomial model. Multivariate normal model.

**Read**: BDA Ch. 3. | Slides

**Code**: Two-parameter Gaussian model | Prediction with two-parameter Gaussian model | Multinomial model

**Math exercises 1**: One-parameter models.

Problem set 1 | Solution Problem 2 and 4 | Solution Problem 1 and 3 (Problem 3 is marked as Problem 2 in this solution)

**Lab 1**: Exploring posterior distributions in one-parameter models by simulation and direct numerical evaluation.

Lab 1 | LISAM Submission

#### Module 2 - Bayesian Regression and Classification

**Lecture 4**: Prediction. Making Decisions.

**Read**: BDA Ch. 9.1-9.2. | Slides

**Lecture 5**: Linear Regression. Nonlinear regression. Regularization priors.

**Read**: BDA Ch. 14 and Ch. 20.1-20.2 | Slides

**Lecture 6**: Classification. Posterior approximation. Logistic regression. Naive Bayes.

**Read**: BDA Ch. 16.1-16.3 | Slides

**Code**: Logistic and Probit Regression

**Math exercises 2**: Predictive distributions and decisions.

Problem set 2 | Solutions

**Lab 2**: Polynomial regression and classification with logistic regression

Lab 2 | Linköping temperature data | Women work data | LISAM Submission

#### Module 3 - More Advanced Models, MCMC and Variational Bayes

**Lecture 7**: Bayesian computations. Monte Carlo simulation. Gibbs sampling. Data augmentation.

**Read**: BDA Ch. 10-11 | Slides

**Code**: Gibbs sampling for a bivariate normal | Gibbs sampling for a mixture of normals

**Lecture 8**: MCMC and Metropolis-Hastings

**Read**: BDA Ch. 11 | Slides

**Code**: Simulating Markov Chains

**Lecture 9**: HMC, Variational Bayes and Stan.

**Read**: BDA Ch. 12.4 and Ch. 13.7 and RStan vignette| Slides

**Code**: RStan - Three Plants | RStan - Bernoulli model | RStan - Logistic regression | RStan - Logistic regression with random effects | RStan - Poisson model

**Math exercises 3**: Comparing Bayes and Frequentist. Posterior approximation. Naive Bayes.

Problem set 3 | Solutions | R-code for exercise 3

**Lab 3**: Gibbs sampling for the normal model, mixture of normals and probit regression.

Lab 3 | Rainfall data | Campylobacter data | LISAM Submission

#### Module 4 - Model Inference and Variable Selection

**Lecture 10**: Bayesian model comparison

**Read**: BDA Ch. 7 | Slides

**Code**: Comparing models for count data

**Lecture 11**: Computing the marginal likelihood, Bayesian variable selection, model averaging.

**Read**: Article on variable selection for additional reading | Slides

**Lecture 12**: Model evaluation and course summary.

**Read**: BDA 6.1-6.4 | Slides

**Math exercises 4**: Model comparison.

Problem set 4 | Solutions to problem set 4

**Lab 4**: Metropolis-Hastings for Poisson regression.

Lab 4 | eBay data | LISAM Submission

#### Literature

**Bayesian Data Analysis**by Gelman, Carlin, Stern, och Rubin, Chapman & Hall, Third edition. The book's web site can be found here.- My
**slides**.

#### Examination

The examination for the course Bayesian Learning, 6 credits, consists of

- Written reports on the four computer labs (3 credits)
- Computer exam (3 credits)

The following material will be made available in the computer exam system in the folder

**given_files**:

- Slides from all the 12 lectures in PDF format
- The four computer labs exercises in PDF format
- You will be able to install any R package from CRAN.
- Four pages with distributions from the Appendix in the course book.
- Some page with useful probability and math results
- Base R cheat sheet and R markdown cheat sheet.
- Your solutions to the four computer labs that you submitted through LISAM (see below).

Here is the LISAM submission for the exam 2018-11-01. Remember to submit before the

**deadline 2018-10-29 at 11.30 AM**.

Here is information about how to log in and out from the exam system: Step by step tutorial for the computer exam system

The exam system includes a so called

**communication client**. This page contains information about the communication client. The communication client serves three basic purposes:

**To ask questions to the teacher**, and for the teacher to inform students about things during the exam. I will also visit the exam room regularly, but do use the communication client to ask questions. To submit a question, just click on the button marked by a dotted blue rectangle in this picture**To get the Client ID**(Klient ID in Swedish). This is the area marked by a dashed red rectangle in this picture. This is your personal identication number that you should write on every page of the papers that you submit for the math questions. Do not write your name on the papers, as that will reveal your identity to the grading teacher.**To submit your (electronic) solutions to the exam**. You will only have a single submission for the whole exam. The exam is submitted by clicking the button marked by a solid green rectangle in this picture and following the instructions. Note that the system will let you know that the exam has been submitted, but will**not**tell you that it was received. This is ok and your solution has actually been received.

- There are three basic folders for the students:
*/home/student_tilde*(home folder where the student can save files)*/home/student_tilde/given_files*(the folder where the distributed files to everyone are placed. For example, the exam itself or slides)*/home/student_tilde/my_given_files*(the folder where the distributed files to individual students are placed. For example, the student's lab solutions)

- The student may have to copy files (e.g. data files) from
*given_files*to their home folder. Or read the files using the full path*/home/student_tilde/given_files*. - Most students will save the plots directly from RStudio using point-and-click from the menus. But you can also save a plot to a png file using these commands:

`png('myPlot.png')`

plot(rnorm(4)) # Change this to the plot commands you want

dev.off()

The course TDDE07 will be graded on the (U,3,4,5) scale.

Here is picture that shows the percentage of the maximum score for each grade (732A91 to the left, TDDE07 to the right).

#### Old exams with solutions

- Exam 2017-05-30 | Solutions: Paper and Code.
- Exam 2017-08-16 | Solutions: Paper and Code.
- Exam 2017-10-27 | Solutions: Paper and Code.
- Exam 2018-06-01 | Solutions: Paper and Code.
- Exam 2018-08-22 | Solutions: Paper and Code.
- Exam 2018-11-01 | Solutions: Paper and Code.

#### R stuff

- The main page with links to downloads for the programming language R
- RStudio - a very nice developing environment for R.
- Short introduction to R | A little longer introduction | John Cook's intro to R for programmers.

#### Other material

- Informative clickable chart with relations between distributions: http://www.johndcook.com/distribution_chart.html.
- Learning about the prior-to-posterior mapping in:
- Bernoulli model with Beta prior

Google Docs | RStudio manipulate - Normal model with normal prior.

Google Docs - Poisson model with Gamma prior

Google Docs - Normal model with Laplace prior.

Google Docs

- Bernoulli model with Beta prior
- The Feynman technique to learning: 5 min Youtube video.

Page responsible: Mattias Villani

Last updated: 2018-11-23