An Introduction to Gaussian Bayesian Networks

In the last article, we talked about networks where we have a mix of both discrete and continuous Random Variables. We saw that we were able to use Gaussian Distributions to describe continuous variables, and in this article, we will continue to talk about so-called Gaussian Networks and how we would want to parameterize them in a manner that is the most intelligent and useful.

Understanding the Multivariate Gaussian Distribution

Since we will consider Gaussian Networks, then we should first familiarize ourselves with Gaussian Distributions. This is since they, literally, be the backbone of our network. However, it is not just any Gaussian Distribution we will need to look closer at — it is, specifically, the Multivariate Gaussian Distribution. We will first try to understand what exactly the Gaussian Distribution is before we move on to its multivariate generalization. The Gaussian Distribution is just another name for the Normal Distribution, which we all know — maybe not mathematically in an intimate manner, but we can easily recognize its shape:

The Gaussian Distribution is given by two parameters: The mean, μ, and the variance, σ². Here, variance is a measurement of how spread out the distribution is, while the mean is simply the middle of the distribution:

We have now understood what is meant by a Gaussian Distribution since it is simply an average normal distribution. However, what exactly is a Multivariate Gaussian Distribution? We can now imagine that instead of only having our Gaussian Distribution going in two directions — along the x-axis and up on the y-axis, now it also goes in a multitude of other directions. In other words, it is our Gaussian Distribution in a higher dimension! Let us try to see an illustration of this concept:

Here, we can clearly still see the general shape of the normal distribution — it has, however, simply been extended to 3 dimensions…