Before you ask a mathematician if they can visualize the fourth dimension, ask them if they can truly visualize a three-dimensional object, like the boundary of a four-dimensional football. If they tell you it’s easy, and their name isn’t Maryna Viazovska, they’re probably lying.
Making an accurate picture of an object from a high dimensional space is very challenging. In this blog post we’ll see a surprising case where it turns out to be possible. We’ll visualize an interesting seven-dimensional object, which comes from a question in statistics.
Let’s consider the probability that each of the teams in the quarter-finals of the Men’s FIFA 2018 World Cup would win. The teams were (Uruguay, France, Brazil, Belgium, Russia, Croatia, Sweden, England). Today we know the probabilities of the teams winning, in that order, are , because France has already won. Back on 3rd July the probabilities (according to FiveThirtyEight) were , and on 7th July the probabilities were .
In a recent project we were studying which probability distributions lie in a particular statistical model. We found out that our statistical model is given by inequalities that the eight probabilities need to satisfy. If we call the probabilities , the inequalities are:
The probabilities have to sum to 1, so . We want to visualize the part of seven-dimensional space in which the inequalities hold. How can we do it?
The first step is to notice that some combinations of letters do not affect whether the inequalities hold or not. They are:
So we can apply a change of coordinates that removes these three directions, leaving something four-dimensional. Finally, to get something three-dimensional we can assume that the four remaining coordinates lie on the sphere.
We end up with a picture that looks like this:
The part of space that lies inside the statistical model are the points outside either the blue blob, the green blob, or the yellow blob.
These days, we have an even better way to visualize the statistical model, truly in 3D. It even doubles-up as a handmade toy for children.
We can’t help but wonder – which other children’s toys are really statistical models in disguise?