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

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.