Continuous and randomized defensive forecasting: unified view
Abstract
Description
Defensive forecasting is a method of transforming laws of probability (stated in game-theoretic terms as strategies for Sceptic) into forecasting algorithms. There are two known varieties of defensive forecasting: "continuous", in which Sceptic's moves are assumed to depend on the forecasts in a (semi)continuous manner and which produces deterministic forecasts, and "randomized", in which the dependence of Sceptic's moves on the forecasts is arbitrary and Forecaster's moves are allowed to be randomized. This note shows that the randomized variety can be obtained from the continuous variety by smearing Sceptic's moves to make them continuous.
10 pages. The new version: (1) relaxes the assumption that the outcome space is finite, and now it is only assumed to be compact; (2) shows that in the case where the outcome space is finite of cardinality C, the randomized forecasts can be chosen concentrated on a finite set of cardinality at most C
10 pages. The new version: (1) relaxes the assumption that the outcome space is finite, and now it is only assumed to be compact; (2) shows that in the case where the outcome space is finite of cardinality C, the randomized forecasts can be chosen concentrated on a finite set of cardinality at most C