How to turn a collection of small building blocks into a versatile tool for solving regression problems.

Even if you have spent some time reading about machine learning, chances are that you have never heard of Gaussian processes. And if you have, rehearsing the basics is always a good way to refresh your memory. With this blog post we want to give an introduction to Gaussian processes and make the mathematical intuition behind them more approachable.

Gaussian processes are a powerful tool in the machine learning toolbox*fitting* a function to the data.
This is called regression and is used, for example, in robotics or time series forecasting.
But Gaussian processes are not limited to regression — they can also be extended to classification and clustering tasks

We will first explore the mathematical foundation that Gaussian procsses are built on — we invite you to follow along using the interactive figures and hands-on examples. They help to explain the impact of individual components, and show the flexibility of Gaussian processes. After following this article we hope that you will have a visual intuition on how Gaussian processes work and how you can configure them for different types of data.

Before we can explore Gaussian processes, we need to understand the mathematical concepts they are based on.
As the name suggests, the Gaussian distribution (which is often also referred to as *normal* distribution) is the basic building block of Gaussian processes.
In particular, we are interested in the multivariate case of this distribution, where each random variable is distributed normally and their joint distribution is also Gaussian.
The multivariate Gaussian distribution is defined by a mean vector

The mean vector

We say

Visually, the distribution is centered arround the mean and the covariance matrix defines its shape. The following figure shows the influence of these parameters on a two-dimensional Gaussian distribution. The standard deviations for each random variable are on the diagonal of the covariance matrix, while the other values show the covariance between them.

Gaussian distributions are widely used to model the real world.
For example, we can employ them to describe errors of measurements or phenomena under the assumptions of the *central limit theorem*

Gaussian distributions have the nice algebraic property of being closed under conditioning and marginalization. Being closed under conditioning and marginalization means that the resulting distributions from these operations are also Gaussian, which makes many problems in statistics and machine learning tractable. In the following we will take a closer look at both of these operations, as they are the foundation for Gaussian processes.

*Marginalization* and *conditioning* both work on subsets of the original distribution and we will use the following notation:

With

Through *marginalization* we can extract partial information from multivariate probability distributions. In particular, given a normal probability distribution

The interpretation of this equation is that each partition

The way to interpret this equation is that if we are interested in the probability density of

Another important operation for Gaussian processes is *conditioning*.
It is used to determine the probability of one variable depending on another variable.
Similar to marginalization, this operation is also closed and yields a modified Gaussian distribution.
This operation is the cornerstone of Gaussian processes since it allows Bayesian inference, which we will talk about in the next section.
Conditioning is defined by:

Note that the new mean only depends on the conditioned variable, while the covariance matrix is independent from this variable.

Now that we have worked through the necessary equations, we will think about how we can understand the two operations visually. While marginalization and conditioning can be applied to multivariate distributions of many dimensions, it makes sense to consider the two-dimensional case as shown in the following figure. Marginalization can be seen as integrating along one of the dimensions of the Gaussian distribution, which is in line with the general definition of the marginal distribution. Conditioning also has a nice geometric interpretation — we can imagine it as making a cut through the multivariate distribution, yielding a new Gaussian distribution with fewer dimensions.

Now that we have recalled some of the basic properties of multivariate Gaussian distributions, we will combine them together to define Gaussian processes, and show how they can be used to tackle regression problems.

First, we will move from the continuous view to the discrete representation of a function:
rather than finding an implicit function, we are interested in predicting the function values at concrete points, which we call *test points*

Now, the goal of Gaussian processes is to learn this underlying distribution from *training data*.
Respective to the test data

In order to perform regression on the training data, we will treat this problem as *Bayesian inference*.
The essential idea of Bayesian inference is to update the current hypothesis as new information becomes available.
In the case of Gaussian processes, this information is the training data.
Thus, we are interested in the conditional probability

Now that we have the basic framework of Gaussian processes together, there is only one thing missing:
how do we set up this distribution and define the mean *covariance function* *kernel* of the Gaussian process.
We will talk about this in detail in the next section.
But before we come to this, let us reflect on how we can use multivariate Gaussian distributions to estimate function values.
The following figure shows an example of this using ten test points at which we want to predict our function:

In Gaussian processes we treat each test point as a random variable.
A multivariate Gaussian distribution has the same number of dimensions as the number of random variables.
Since we want to predict the function values at

Recall that in order to set up our distribution, we need to define *centering* of the data.
So configuring

The clever step of Gaussian processes is how we set up the covariance matrix *covariance function*, pairwise on all the points.
The kernel receives two points

We evaluate this function for each pairwise combination of the test points to retrieve the covariance matrix.
This step is also depicted in the figure above.
In order to get a better intution for the role of the kernel, let's think about what the entries in the covariance matrix describe.
The entry

Kernels are widely used in machine learning, for example in *support vector machines*

Kernels can be separated into *stationary* and *non-stationary* kernels. *Stationary* kernels, such
as the RBF kernel, are functions invariant to translations, and the covariance of two points is only
dependent on their relative position. A special case here would be the periodic kernel, which is only invariant to translations equal to the period of its respective function. *Non-stationary* kernels, such as the linear kernel, do not have this
constraint and depend on an absolute location. The stationary nature of the RBF kernel can be observed in the
banding around the diagonal of its covariance matrix (as shown in this figure). Increasing the length parameter increases the banding, as
points further away from each other become more correlated. For the periodic kernel, we have an additional parameter

There are many more kernels that can describe different classes of functions, which can be used to model the desired shape of the function.
A good overview of different kernels is given by Duvenaud

We will now shift our focus back to the original task of regression.
As we have mentioned earlier, Gaussian processes define a probability distribution over possible functions.
In this figure above, we show this connection:
each sample of our multivariate normal distribution represents one realization of our function values.
Because this distribution is a multivariate Gaussian distribution, the distribution of functions is normal.
Recall that we usually assume *prior* distribution

If we have not yet observed any training examples, this distribution revolves around

In the previous section we have looked at examples of different kernels. The kernel is used to define the entries of the covariance matrix. Consequently, the covariance matrix determines which type of functions from the space of all possible functions are more probable. As the prior distribution does not yet contain any additional information, it is perfect to visualize the influence of the kernel on the distribution of functions. The following figure shows samples of potential functions from prior distributions that were created using different kernels:

Adjusting the parameters allows you to control the shape of the resulting functions.
This also varies the confidence of the prediction.
When decreasing the variance *Linear* kernel, setting the variance

So what happens if we observe training data?
Let's get back to the model of Bayesian inference, which states that we can incorporate this additional information into our model, yielding the *posterior* distribution

First, we form the joint distribution

For the next step we need one operation on Gaussian distributions that we have defined earlier.
Using *conditioning* we can find

As mentioned before, the conditional distribution

We do this by slightly modifying the setup of the joint distribution

Again, we can use conditioning to derive the predictive distribution

Analogous to the prior distribution, we could obtain a prediction for our function values by sampling from this distribution. But, since sampling involves randomness, the resulting fit to the data would not be deterministic and our prediction could end up being an outlier. In order to make a more meaningful prediction we can use the other basic operation of Gaussian distributions.

Through *marginalization* of each random variable, we can extract the respective mean function value

The following figure shows an example of the conditional distribution.
At first, no training points have been observed.
Accordingly, the mean prediction remains at

The training points can be activated by clicking on them, which leads to a constrained distribution. This change is reflected in the entries of the covariance matrix, and leads to an adjustment of the mean and the standard deviation of the predicted function. As we would expect, the uncertainty of the prediction is small in regions close to the training data and grows as we move further away from those points.

In the constrained covariance matrix, we can see that the correlation of neighbouring points is affected by the training data. If a predicted point lies on the training data, there is no correlation with other points. Therefore, the function must pass directly through it. Predicted values further away are also affected by the training data — proportional to their distance.

As described earlier, the power of Gaussian processes lies in the choice of the kernel function. This property allows experts to introduce domain knowledge into the process and lends Gaussian processes their flexibility to capture trends in the training data. For example, by choosing a suitable bandwidth for the RBF kernel, we can control how smooth the resulting function will be.

A big benefit that kernels provide is that they can be combined together, resulting in a more specialized kernel.
The decision which kernel to use is highly dependent on prior knowledge about the data, e.g. if certain characteristics are expected.
Examples for this would be stationary nature, or global trends and patterns.
As introduced in the section on kernels, stationary means that a kernel is translation invariant and therefore not dependent on the index

Let's consider two kernels, a linear kernel

However, combinations are not limited to the above example, and there are more possibilities such as concatenation or composition with a function

Knowing more about how kernel combinations influence the shape of the resulting distribution, we can move on to a more complex example.
In the figure below, the observed training data has an ascending trend with a periodic deviation.
Using only a linear kernel, we can mimic a normal linear regression of the points.
At first glance the RBF kernel accurately approximates the points.
But since the RBF kernel is stationary it will always return to

As discussed in the section about GPs, a Gaussian process can model uncertain observations. This can be seen when only selecting the linear kernel, as it allows us to perform linear regression even if more than two points have been observed, and not all functions have to pass directly through the observed training data.

With this article, you should have obtained an overview of Gaussian processes, and developed a deeper understanding on how they work. As we have seen, Gaussian processes offer a flexible framework for regression and several extensions exist that make them even more versatile.

For instance, sometimes it might not be possible to describe the kernel in simple terms.
To overcome this challenge, learning specialized kernel functions from the underlying data, for example by using deep learning*model-peeling* and hypothesis testing.
By comparing different kernels on the dataset, domain experts can introduce additional knowledge through
appropriate combination and parameterization of the kernel.

If we have sparked your interest, we have compiled a list of further blog posts on the topic of Gaussian processes. In addition, we have linked two Python notebooks that will give you some hands-on experience and help you to get started right away.

We are very grateful to Carla Avolio and Marc Spicker for their feedback on the manuscript. In addition, we want to thank Jonas Körner for helping with the implementation of the figure explaining the multivariate Gaussian distribution. Furthermore, we would like to thank the German Research Foundation (DFG) for financial support within project A01 of the SFB-TRR 161 and within the Research Unit FOR 2111 with grant number DFG-431/16.

The following blog posts offer more interactive visualizations and further reading material on the topic of Gaussian processes:

- Gaussian process regression demo by Tomi Peltola
- Gaussian Processes for Dummies by Katherine Bailey
- Intuition behind Gaussian Processes by Mike McCourt
- Fitting Gaussian Process Models in Python by Chris Fonnesbeck

If you want more of a hands-on experience, there are also many Python notebooks available:

- Fitting Gaussian Process Models in Python by Chris Fonnesbeck
- Gaussian process lecture by Andreas Damianou