2,068 research outputs found
Continuous and randomized defensive forecasting: unified view
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.Comment: 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
Competitive on-line learning with a convex loss function
We consider the problem of sequential decision making under uncertainty in
which the loss caused by a decision depends on the following binary
observation. In competitive on-line learning, the goal is to design decision
algorithms that are almost as good as the best decision rules in a wide
benchmark class, without making any assumptions about the way the observations
are generated. However, standard algorithms in this area can only deal with
finite-dimensional (often countable) benchmark classes. In this paper we give
similar results for decision rules ranging over an arbitrary reproducing kernel
Hilbert space. For example, it is shown that for a wide class of loss functions
(including the standard square, absolute, and log loss functions) the average
loss of the master algorithm, over the first observations, does not exceed
the average loss of the best decision rule with a bounded norm plus
. Our proof technique is very different from the standard ones and
is based on recent results about defensive forecasting. Given the probabilities
produced by a defensive forecasting algorithm, which are known to be well
calibrated and to have good resolution in the long run, we use the expected
loss minimization principle to find a suitable decision.Comment: 26 page
On a simple strategy weakly forcing the strong law of large numbers in the bounded forecasting game
In the framework of the game-theoretic probability of Shafer and Vovk (2001)
it is of basic importance to construct an explicit strategy weakly forcing the
strong law of large numbers (SLLN) in the bounded forecasting game. We present
a simple finite-memory strategy based on the past average of Reality's moves,
which weakly forces the strong law of large numbers with the convergence rate
of . Our proof is very simple compared to a corresponding
measure-theoretic result of Azuma (1967) on bounded martingale differences and
this illustrates effectiveness of game-theoretic approach. We also discuss
one-sided protocols and extension of results to linear protocols in general
dimension.Comment: 14 page
Hoeffding's inequality in game-theoretic probability
This note makes the obvious observation that Hoeffding's original proof of
his inequality remains valid in the game-theoretic framework. All details are
spelled out for the convenience of future reference.Comment: 5 page
Defensive forecasting for optimal prediction with expert advice
The method of defensive forecasting is applied to the problem of prediction
with expert advice for binary outcomes. It turns out that defensive forecasting
is not only competitive with the Aggregating Algorithm but also handles the
case of "second-guessing" experts, whose advice depends on the learner's
prediction; this paper assumes that the dependence on the learner's prediction
is continuous.Comment: 14 page
Defensive online portfolio selection
The class of defensive online portfolio selection algorithms,designed for fi nite investment horizon, is introduced. The Game Constantly Rebalanced Portfolio and the Worst Case Game Constantly Rebalanced Portfolio, are presented and theoretically analyzed. The analysis exploits the rich set of mathematical tools available by means of the connection between Universal Portfolios and the Game- Theoretic framework. The empirical performance of the Worst Case Game Constantly Rebalanced Portfolio algorithm is analyzed through numerical experiments concerning the FTSE 100, Nikkei 225, Nasdaq 100 and S&P500 stock markets for the time interval, from January 2007 to December 2009, which includes the credit crunch crisis from September 2008 to March 2009. The results emphasize the relevance of the proposed online investment algorithm which signi fi cantly outperformed the market index and the minimum variance Sharpe-Markowitz’s portfolio.on-line portfolio selection; universal portfolio; defensive strategy
Implications of contrarian and one-sided strategies for the fair-coin game
We derive some results on contrarian and one-sided strategies by Skeptic for
the fair-coin game in the framework of the game-theoretic probability of Shafer
and Vovk \cite{sv}. In particular, concerning the rate of convergence of the
strong law of large numbers (SLLN), we prove that Skeptic can force that the
convergence has to be slower than or equal to . This is achieved
by a very simple contrarian strategy of Skeptic. This type of result, bounding
the rate of convergence from below, contrasts with more standard results of
bounding the rate of SLLN from above by using momentum strategies. We also
derive a corresponding one-sided result
A betting interpretation for probabilities and Dempster-Shafer degrees of belief
There are at least two ways to interpret numerical degrees of belief in terms
of betting: (1) you can offer to bet at the odds defined by the degrees of
belief, or (2) you can judge that a strategy for taking advantage of such
betting offers will not multiply the capital it risks by a large factor. Both
interpretations can be applied to ordinary additive probabilities and used to
justify updating by conditioning. Only the second can be applied to
Dempster-Shafer degrees of belief and used to justify Dempster's rule of
combination.Comment: 20 page
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