Confidence Predictor

A confidence predictor is essentially a prediction algorithm producing prediction regions. In the context of conformal prediction, we assume that Reality outputs successive pairs \[ (x_1, y_1), (x_2, y_2), \ldots\] called observations. The objects ⚠ $x_i$ are elements of a measurable space ⚠ $\mathbf{X}$ and the labels ⚠ $y_i$ are elements of a measurable space ⚠ $\mathbf{Y}$.

We call ⚠ $\mathbf{Z}:=\mathbf{X}\times\mathbf{Y}$ the observation space, ⚠ $\epsilon\in(0,1)$ the significance level, and the complimentary value ⚠ $1 - \epsilon$ the confidence level.

A confidence predictor is a measurable function ⚠ $\Gamma: \mathbf{Z}^*\times \mathbf{X}\times (0,1)\to 2^Y$ (⚠ $2^{\mathbf{Y}}$ is a set of all subsets of ⚠ $\mathbf{Y}$) that satisfies \[ \Gamma^{\epsilon_1}(x_1, y_1, \ldots, x_n, y_n, x_{n+1}) \subseteq \Gamma^{\epsilon_2}(x_1, y_1, \ldots, x_n, y_n, x_{n+1})\] for all significance levels ⚠ $\epsilon_1\ge\epsilon_2$, all positive integers ⚠ $n$, and all incomplete data sequences ⚠ $x_1, y_1, \ldots, x_n, y_n, x_{n+1}$. Thus, a confidence predictor is an algorithm that given an incomplete data sequence and ⚠ $\epsilon \in (0,1)$ (the significance level), outputs a subset ⚠ $ \Gamma^\epsilon(x_1, y_1, \ldots, x_n, y_n, x_{n+1})$ of ⚠ $\mathbf{Y}$ (the prediction region) so that the condition above is satisfied.