Notation and major confusion with the definition of the probability simplex

Question

So it starts like this:

Given a discrete set $N$, the probability simplex over $N$, denoted $∆(N)$ is defined to be:

$$Δ(N) = \left\{ x \in \mathbb{R}^{|N|} \;:\; x_{i} \geq 0 \text{ for all } i, \text{ and } \sum_{i=1}^{|N|} x_i = 1\right\}$$

I guess I kind of understand what it tries to convey, but in practice something doesn't add up.

For example, suppose, that we have $x= \{x_1,x_2\}$ with dimentionality $|N| = 3$ according to the specification above.

Say:

that $x_1 = \{\mathbb{R}_1, \mathbb{R}_2, \mathbb{R}_3\}$ and $x_2 = \{\mathbb{R}_4, \mathbb{R}_5, \mathbb{R}_6\}$.

How does this summation $\sum_{i=1}^{|N|} x_i = 1$ work?

Answer 1

I think, this is a kind of question that should be answered via examples.

For any set $N_2=\{a,b\}$ with 2 elements the following vectors $x_1, x_2, x_3$ belong to $\Delta(N_2)$:

$$ x_1 = \begin{bmatrix} 0.2 \\ 0.8 \end{bmatrix}, \quad x_2 = \begin{bmatrix} 0.4 \\ 0.6 \end{bmatrix}, \quad x_3 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}. $$ For each of them we have $x_i \in \Delta(N_2)$. Of course there are many more such vectors.

For any set $N_3=\{a,b,c\}$ with 3 elements the following vectors $x_1, x_2, x_3$ belong to $\Delta(N_3)$:

$$ x_1 = \begin{bmatrix} 0.2 \\ 0.3 \\ 0.5 \end{bmatrix}, \quad x_2 = \begin{bmatrix} 0.4 \\ 0.4 \\ 0.2 \end{bmatrix}, \quad x_3 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}. $$ For each of them we have $x_i \in \Delta(N_3)$. Again, there are many more elements of $\Delta(N_3)$.

Note that the set $N$ is actually of no matter for constructing $\Delta(N)$, it is only its size $|N|$ that matters.

Answer 2

Well, what is meant under $\triangle(N)$ is the following: if your vector $x \in \mathbf{R}^{|N|}$ has the following properties:

1. $x_i \ge 0$ for all $i$

2. $\sum_{i=1}^{|N|} x_i = 1$

Then, your vector is from simplex $\triangle(N)$. The sum is taken over all coordinates of the vector.

Clearly, if one of these conditions is violated then your vector doesn't belong to simplex $\triangle(N)$.