Foundations
These are some of the known facts about game-theoretic upper probability ⚠ $P$
:
- It is an outer measure [obvious].
- It is a Choquet capacity, at least in the case of a finite outcome space and one-step ahead forecasts [Vovk 2009].
- In general, it is not strongly additive, i.e., it is not guaranteed to satisfy
⚠ $P(A\cup B) + P(A\cap B) \le P(A) + P(B)$
. (Therefore, the situation is similar to that in the theory of imprecise probabilities: cf. Walley (2000), page 128.) This is a simple example in the prequential framework (sequential probability forecasting of binary outcomes) with horizon 2 (i.e., the forecaster issues 2 forecasts⚠ $p_1,p_2$
for 2 consecutive outcomes⚠ $y_1,y_2$
):⚠ $A = \{(0,0,1/2,0),(1/2,0,0,0)\}$
and⚠ $B = \{(0,0,1/2,0),(1/2,1,0,0)\}$
(the elements of⚠ $A$
and⚠ $B$
are represented in the form⚠ $(p_1,y_1,p_2,y_2)$
). In this case we have⚠ $P(A\cup B)=1$
and⚠ $P(A\cap B) = P(A) = P(B) = 1/2$
. (For the standard prequential framework with infinite horizon just add⚠ $00\ldots$
at the end of each element of⚠ $A$
and⚠ $B$
.)
Bibliography
- Vladimir Vovk. Prequential probability: game-theoretic = measure-theoretic. The Game-Theoretic Probability and Finance Project,Working Paper 27, January 2009.
- Peter Walley. Towards a unified theory of imprecise probability. International Journal of Approximate Reasoning 24 (2000) 125 - 148.