# Week 5

# Bayesian Regression

Bayesian Inference Lecture Slides.

## Lab Class

The notebook for the lab class can be downloaded from here.

To obtain the lab class in ipython notebook, first open the ipython notebook. Then paste the following code into the ipython notebook

```
import urllib
urllib.urlretrieve('https://github.com/SheffieldML/notebook/blob/master/lab_classes/machine_learning/MLAI_lab4.ipynb', 'MLAI_lab4.ipynb')
```

You should now be able to find the lab class by clicking `File->Open`

on
the ipython notebook menu.

### YouTube Videos

There is a YouTube video available of me giving this material at the Gaussian Process Road Show in Uganda.

#### GPRS Uganda Video

Second half overlaps with the material from this week’s lectures.

#### Video from 2011 on Gaussian Densities and Bayesian Inference

### Reading

- Rogers and Girolami Chapter 3: Bayesian Methods Section 3.1-3.3 (pg 95-117)
- Sections 1.2.3 (pg 21-24) of Bishop
- Sections 1.2.6 (start from just past equ 1.64, pg 30-32) of Bishop
- Section 2.3 of Bishop up to top of pg 85 (multivariate Gaussians).
- Section 3.3 of Bishop up to pg 159 (pg 152-159). (Bayesian linear regression)
- Sections 3.7-3.8 of Rogers and Girolami (pg 122-133).
- Section 3.4 of Bishop (pg 161-165).

### Previous Lectures

Univariate Bayesian Inference

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Multivariate Bayesian Inference

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Bayesian Polynomials on Olympics Data

## Learning Outcomes Week 5

- Understand the principal of integrating parameters and how to use Bayes rule to do so.
- Understand the role of the prior distribution.
- In multivariate and univariate Gaussian examples, be able to combine the prior with the likelihood to form a posterior distribution..
- Recognise the role of the marginal likelihood and know its form for Bayesian regression under Gaussian priors.
- Be able to compute the expected output of the model and its covariance using the posterior distribution and the formula for the function.
- Understand the effect of model averaging and its advantages when
making predictions including:
- Error bars
- Regularized prediction (reduces variance)