Recently, I came across the following image of a "probability distribution" defined over a "simplex":
Question: In the above picture, what exactly is the "simplex"?
I tried reading about the official definition of a "simplex":
However, I can't seem to understand the relationship between the "simplex" in which the probability distribution is defined compared to the official definition of the "simplex".
Can someone please help me understand what exactly is a "simplex" in this context? Does this "simplex" have any relationship to the "simplex method" (https://en.wikipedia.org/wiki/Simplex_algorithm) in optimization and linear programming?
References:
There are other notions of simplex, from topology in particular. We’ll only cover affine simplices here.
In general, an $n$-simplex is the convex hull of $n+1$ points such that no three points are colinear, no four points are coplanar, and more generally, no $k+1$ points are in the same $k-1$-dimensional affine space.
Given points $p_0,\cdots,p_{n},$ the points in the simplex can be written uniquely as:
$$a_0p_0+\cdots+a_np_n$$
Where the $a_i$ are non-negative real numbers and $\sum a_i=1.$
This means that a common primary representative of an $n$-simplex is: $$\Delta_n=\left\{(a_0,\dots,a_n)\in\mathbb R^{n+1}\mid a_i\geq 0, \sum a_i=1\right\}$$ This is the context hull of the points in $\mathbb R^{n+1}$ of the $n+1$ points:$$e_0=(1,0,0,\dots,0),\\e_1=(0,1,0,\dots,0),\\\vdots\\,e_n=(0,0,0,\dots,1)$$ $\Delta_n$ has a nice geometric symmetry. When $n=2,$ it is an equilateral triangle, when $n=3,$ title is a regular tetrahedron. In general, it is an example of a regular $n$-dimensional “solid.”
However, this $\Delta_n$ is only one simplex. It’s main disadvantage is that it represents an $n$-dimensional object in $n+1$-dimensions.
In the above diagram, the three points are $(0,0),$ $(1,0)$ and $(0,1)$ and the simplex consists of all $(x,y)$ with $0\leq x,y$ and $x+y\leq 1.$
That is part of another representative $n$-simplex, but in $\mathbb R^n,$ of all $(a_1,\dots,a_n)$ with $a_i\geq 0$ and $\sum a_i\leq 1.$