Classification Problem

Last updated Nov 1, 2022

Let $(\Omega, \mathcal{M}, \mathbb{P})$ be a Probability Space. Let $(\mathcal{D}, \Sigma_{1})$ and $([n], \mathcal{P}([n]))$ be Measure Spaces (for $n \in \mathbb{N}$). Let $X: \Omega \to \mathcal{D}$ and $Y: \Omega \to [n]$ be Random Elements representing Data and Labels respectively. A Classification Problem is a Tuple $(\Omega, \mathcal{M}, \mathbb{P}, \mathcal{D}, \Sigma_{1}, n, X, Y)$.

1. The goal of this problem is to find a Function $f: \mathcal{D} \to [n]$ so that the Risk of $f$ is minimized. We say $f$ is a Classifier
2. This definition is a bit unwieldy. We usually just refer to the problem as $(X, Y, n)$ or $(X, Y)$ instantiating all other Objects implicitly.