# How is the set of probability distributions on $m$ values an $m-1$-dimensional simplex? [duplicate]

In Zuk et al. 2012, they claim:

The set of probability distributions on $$m$$ values is the $$m-1$$-dimensional simplex denoted $$S_m$$.

Probability distribution functions are familiar to me. I understand the basic notion of a simplex as a geometric object, and I recall them from how the simplex method can be geometrically interpreted as a traversal of simplex nodes. But I am not familiar with representing sets of probability distributions as a simplex.

How does this representation work?

$$\left\{x \in \mathbb{R}^{k} : x_0 + \dots + x_{k-1} = 1, x_i \ge 0 \text{ for } i = 0, \dots, k-1 \right\}$$
It’s the set of $$k$$-dimensional points whose coordinates sum to 1 and are all non-negative. This lines up with the axioms of probability. The notion becomes clearer if you just replace each $$x$$ with a $$p$$.
As far as why it’s the $$m-1$$-simplex instead of $$m$$, consider the example of a distribution over three classes. The coordinates are points like $$(0.3, 0.1, 0.6)$$. These exist in a 3D space, but really they exist in a 2D subspace—a triangle whose vertices are at $$(0,0,1)$$, $$(0,1,0)$$, and $$(1,0,0)$$.