782 research outputs found
Jeffreys's law for general games of prediction: in search of a theory
We are interested in the following version of Jeffreys's law: if two
predictors are predicting the same sequence of events and either is doing a
satisfactory job, they will make similar predictions in the long run. We give a
classification of instances of Jeffreys's law, illustrated with examples.Comment: 12 page
Leading strategies in competitive on-line prediction
We start from a simple asymptotic result for the problem of on-line
regression with the quadratic loss function: the class of continuous
limited-memory prediction strategies admits a "leading prediction strategy",
which not only asymptotically performs at least as well as any continuous
limited-memory strategy but also satisfies the property that the excess loss of
any continuous limited-memory strategy is determined by how closely it imitates
the leading strategy. More specifically, for any class of prediction strategies
constituting a reproducing kernel Hilbert space we construct a leading
strategy, in the sense that the loss of any prediction strategy whose norm is
not too large is determined by how closely it imitates the leading strategy.
This result is extended to the loss functions given by Bregman divergences and
by strictly proper scoring rules.Comment: 20 pages; a conference version is to appear in the ALT'2006
proceeding
Competing with stationary prediction strategies
In this paper we introduce the class of stationary prediction strategies and
construct a prediction algorithm that asymptotically performs as well as the
best continuous stationary strategy. We make mild compactness assumptions but
no stochastic assumptions about the environment. In particular, no assumption
of stationarity is made about the environment, and the stationarity of the
considered strategies only means that they do not depend explicitly on time; we
argue that it is natural to consider only stationary strategies even for highly
non-stationary environments.Comment: 20 page
Game-theoretic versions of strong law of large numbers for unbounded variables
We consider strong law of large numbers (SLLN) in the framework of
game-theoretic probability of Shafer and Vovk (2001). We prove several versions
of SLLN for the case that Reality's moves are unbounded. Our game-theoretic
versions of SLLN largely correspond to standard measure-theoretic results.
However game-theoretic proofs are different from measure-theoretic ones in the
explicit consideration of various hedges. In measure-theoretic proofs existence
of moments are assumed, whereas in our game-theoretic proofs we assume
availability of various hedges to Skeptic for finite prices
Rough paths in idealized financial markets
This paper considers possible price paths of a financial security in an
idealized market. Its main result is that the variation index of typical price
paths is at most 2, in this sense, typical price paths are not rougher than
typical paths of Brownian motion. We do not make any stochastic assumptions and
only assume that the price path is positive and right-continuous. The
qualification "typical" means that there is a trading strategy (constructed
explicitly in the proof) that risks only one monetary unit but brings infinite
capital when the variation index of the realized price path exceeds 2. The
paper also reviews some known results for continuous price paths and lists
several open problems.Comment: 21 pages, this version adds (in Appendix C) a reference to new
results in the foundations of game-theoretic probability based on Hardin and
Taylor's work on hat puzzle
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