How to visualize Dirichlet distribution (with more than 3 targets)?

Question

I want to plot a Dirichlet distribution $\operatorname{Dir}(\alpha), \alpha=[\alpha_1, \alpha_2, \ldots,\alpha_n]$. However, when I google it, almost all of the results consider 3 targets ($n=3$), and the distribution can be visualized by a triangle (e.g. Dirichlet distribution from Wikipedia).

What if I have 4 ($n=4$) or 5 ($n=5$) targets? Will the visualization be rectangle or pentagon?

To obtain some toy data, we can do

from scipy.stats import dirichlet
import numpy as np

n_samples = 100
alpha = np.array([0.4, 5, 15, 3, 2])
toy_data = dirichlet.rvs(alpha, size=n_samples) #shape (100, 5)

Answer 1

A Dirichlet distribution is a distribution over a simplex.

A simplex is a lower-dimensional ($n-1)$ subspace of an $n$-dimensional space. With two targets, the simplex is a line. With three targets, it’s a triangle. In four, it’s a tetrahedron (not a square)…and so on.

The reason you don’t see 4-target examples is because it’s hard to display probability density on the 3D structure. The reason you don’t see 5-target or higher is that we’re bad at visualizing 4D spaces. Our visual system (and indeed, our entire grip on reality) is centered on 3D space. Trying to visualize more than that is the challenge of Abbott’s famous book Flatland.