The probability simplex $\Delta_K$ represents the set of all possible valid [[Probability Mass Function|PMFs]] over $K$ outcomes. Each point inside the simplex corresponds to one specific combination of probabilities $\mathbf p = [p_1, p_2, \dots, p_K]^T$, satisfying the [[Probability Axioms]].
- *Normalization:*
$ \sum_{j=1}^k p_j=1$
- *Non-negativity:*
$ 0 \le p_j \le 1 \quad \forall j \in \{1,\dots,K\}$
**Examples:**
- For $K=2$, the simplex $\Delta_2$ is a (1-dimensional) line from point $[1,0]$ to point $[0,1]$.
![[probability-simplex.png|center|350]]
- For $K=3$, the simplex $\Delta_3$ is a (2-dimensional) plane forming a triangle.
![[probability-simplex-3d.png|center|400]]