Game Of Prediction

A game of prediction is a triple ⚠ $(\Omega,\Gamma,\lambda)$, where ⚠ $\Omega$ is an outcome space, ⚠ $\Gamma$ is a decision space, and ⚠ $\lambda:\Omega\times\Gamma\to[-\infty,\infty]$ is the loss function. Important special cases are those of:

• point predictions, where ⚠ $\Gamma=\Omega$;
• probability predictions, where ⚠ $\Gamma$ is the set of all probability measures on ⚠ $\Omega$; if ⚠ $\Omega$ is finite, the usual loss function is the log loss;
• region predictions, where ⚠ $\Gamma$ is the set of all subsets of ⚠ $\Omega$; the usual loss function is

⚠ $$ \lambda(\omega,\gamma)= \begin{cases} 1 & \text{if } \omega\in\gamma
0 & \text{otherwise}.\end{cases}⚠ $$

The case of region predictions (in particular, interval predictions) is the main object of study in conformal prediction but has never been studied in competitive on-line prediction. This is stated as open problem in the article competitive on-line interval prediction.