Introduction to Online Convex Optimization Notation

Definition Categorical r.v.

• $\Delta_{n}$,
  • $n - 1$-simplex
• $x \in \Delta_{n}$,
• \(I \in \{1, \ldots, n\}\),
  • r.v.

$I$ is said to categorical r.v. with probability $x$ if its p.d.f. is

\[i = 1, \ldots, n, \ \mathrm{Categ}(i; x) := \sum_{j=1}^{n} x_{j} 1_{\{j = i\}} .\]

■

Definition independent categorical r.v.s over simplex

• $\Delta_{n}$,
  • $n - 1$-simplex

Let $\mathcal{I}$ be a set of independent categorical r.v. with distribution $x \in \Delta_{n}$. That is,

\[I_{1}, \ldots, I_{k} \in \mathcal{I}, \ P(I_{1}, \ldots, I_{k}) = \prod_{i=1}^{k} P(I_{i}) .\]

\(\mathcal{I}_{\Delta_{n}}\) is called independent categorical r.v.s over $\Delta_{n}$.

■