## Tutorial 4: Bayes Point Machine

This tutorial
demonstrates how to use Infer.NET in a standard supervised learning setting.
We will define a classifier, train it using a training set and then apply it to
a test set. The classifier in question is the
Bayes Point Machine (R. Herbrich, T. Graepel, and C. Campbell, **Bayes
Point Machines**, JMLR, 2001) trained via
Expectation Propagation (T. Minka, UAI, 2001).

You can run the code in
this tutorial either using the
Examples Browser or by opening the Tutorials solution in Visual Studio
and executing **BayesPointMachineExample.cs**.

### Will they, won't they?

The problem that we
will look at is trying to predict whether someone will buy a particular product,
based on some demographic information about them. In particular, for each
person, we know their **age** in years and **income **
in thousands of pounds. For the training set, we also know whether or not
they bought the product. Here is the training data:

```
double[] incomes = { 63, 16, 28, 55, 22, 20 };
``` |

We'll use the data to
train our Bayes Point Machine. To do this, we need to create two arrays of
observations, one called **x** consisting of vectors of the input
features (augmented by appending a 1) and one called **y** which
just wraps **willBuy**.

```
// Create x vector, augmented by 1
``` |

### Defining and training the Bayes Point Machine

A Bayes Point Machine
(BPM) classifies an input vector **x** by:

- finding the inner product of
**x**with a weight vector**w** - giving the output
**y**as true if the inner product**w.x**(plus some Gaussian noise) is positive and false otherwise.

When learning a BPM, we
aim to learn a posterior distribution over the weight **w**, given
some prior and the training data. The uncertainty in **w**
will lead to appropriate uncertainty in the predictions given by the BPM on test
data. The noise is essential as it allows for the case where there is no setting
of **w** that perfectly classifies the training data i.e. where the
training data is not linearly separable into the two classes.

We can define a Bayes Point Machine in Infer.NET as follows,

```
Variable<Vector>
w = Variable.Random(
``` |

The first line creates a random variable for **w **with a broad
prior. We then create a range **j** which is the size of the
data. The last line constrains the target **y[j]** to be true if
the inner product **w.x** plus noise is positive or false
otherwise. Note: here we have picked the noise variance to be 0.1 which
sets the 'slack' in the classifier - you should tune this value when using a BPM
for your own problem.

We can now train our Bayes Point Machine by using expectation propagation to
infer the posterior over **w**,

```
InferenceEngine
engine = new
InferenceEngine(new
ExpectationPropagation());
``` |

When this code is executed, the result is

```
Dist over
w=
``` |

This is the multivariate Gaussian approximation to the posterior over our weight vector. Next, we will look at how to use it to classifiy some test data.

### Testing the Bayes Point Machine

We've done all this work so that we can classify new, test examples. In
fact, with a minor modification, we can re-use the existing code to do this.
We put the modelling code in a method called **BayesPointMachine() **
and pass in the variables **w** and **y** along with
the data. The new method looks like this:

```
public void
BayesPointMachine(
``` |

This method can be used for training by passing in **w** and **
y** defined above. We can also use it to classify some test data with the
following:

```
double[] incomesTest
= { 58, 18, 22 };
``` |

where **
wposterior **is the posterior distribution over **w** that
we learned during training. This code prints out

`classifier output=` |

so the BPM classifier has predicted that the first person will almost definitely buy and the other two most likely won't.

I predict you're ready to move on to the next tutorial.