| Abstract: |
Consider the following problem: at each date in the future, a given event may
or may not occur, and you will be asked to forecast, at each date, the
probability that the event will occur in the next date. Unless you make
degenerate forecasts (zero or one), the fact that the event does or does not
occur does not prove your forecast wrong. But, in the long run, if your
forecasts are accurate, the conditional relative frequencies of occurrence of
the event should approach your forecast. [4] has presented an algorithm that,
whatever the sequence of realizations of the event, will meet the long-run
accuracy criterion, even though it is completely ignorant about the real
probabilities of occurrence of the event, or about the reasons why the event
occurs or fails to occur. It is an adaptive algorithm, that reacts to the
history of forecasts and occurrences, but does not learn from the history
anything about the future: indeed, the past need not say anything about the
future realizations of the event. The algorithm only looks at its own past
inaccuracies and tries to make up for them in the future. The amazing result
is that this (making up for past inaccuracies) can be done with arbitrarily
high probability! Alternative arguments for this result have been proposed in
the literature, remarkably by [3], where a very simple algorithm has been
proved to work, using a classical result in game theory: Blackwell’s
approachability result, [1]. Very recently, [2] has especialized Blackwell’s
theorem in a way that (under a minor modification of the algorithm) simplifies
the argument of [3]. Here I present such modification and argument. |