Probability simplex: clarifying dimensions.

Question

I have some doubts about the notion of probability simplex. Consider the region $$ \{(x_1,...,x_K): \sum_{k=1}^K x_k=1 , x_k\geq 0\text{ } \forall k=1,...,K\} $$

1) Is this called a $(K-1)$-probability simplex?

2) Let $K=2$. Then, the graphical representation of the $1$-probability simplex is in the picture below. It is a line segment. Correct?

3) Let $K=3$.Then, I guess that the graphical representation of the $2$-probability simplex is in the picture below. It is a tetrahedron. Correct?

4) If all the above is correct, I'm confused on why we are going from a line segment to a 3-D object. In fact, if I read here, then the 2-D simplex is a triangle.

Could you help me to clarify?

Answer 1

Your guess that the $2$-probability simplex is the whole tetrahedron is not correct. The set of points $\ (x,y,z)\ $ such that $\ x+y+z=1\ $, and $\ 0\le x$, $0\le y, 0\le z\ $, is just the single plane face of the tetrahedron passing through the points $\ (1,0,0),\ (0,1,0),$ and $\ (0,0,1)\ $, that is, a 2-dimensional object.

Answer 2

I'll summarise my thoughts below.

1) I only encountered these in the setting of topology: In which setting it would be called "the $(K-1)$-standard simplex."

N.B. As for the rest, you should note that the $(K-1)$ is referring to the dimension, i.e. "$K$ variables and one constraint".

2) You're absolutely right!

3) This is not correct! The angle you chose to draw the picture I think has mislead you. You're right in that the dimension, ($K-1$) should be equal to $2$!

4) Your confusion is understandable: you should convince yourself that it is the triangle with vertices $(0,0,1)$, $(0,1,0)$, and $(0,0,1)$!

Indeed, you may notice for all $n \ge 1$ that each of the "faces" of the $n$-simplex are in fact copies of the $n-1$ simplex!

Answer 3

To find out the probability simplex you should think of the convex hull, which is the set of all convex combinations of points. Now for K=3, think of convex combination of e1, e2, and e3. If we think of it from the origin (0, 0, 0) we know that it forms a triangle, just 1 face of tetrahedron, just like the hypotenuse of a right triangle.