205 research outputs found
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
A tutorial on conformal prediction
Conformal prediction uses past experience to determine precise levels of
confidence in new predictions. Given an error probability , together
with a method that makes a prediction of a label , it produces a
set of labels, typically containing , that also contains with
probability . Conformal prediction can be applied to any method for
producing : a nearest-neighbor method, a support-vector machine, ridge
regression, etc.
Conformal prediction is designed for an on-line setting in which labels are
predicted successively, each one being revealed before the next is predicted.
The most novel and valuable feature of conformal prediction is that if the
successive examples are sampled independently from the same distribution, then
the successive predictions will be right of the time, even though
they are based on an accumulating dataset rather than on independent datasets.
In addition to the model under which successive examples are sampled
independently, other on-line compression models can also use conformal
prediction. The widely used Gaussian linear model is one of these.
This tutorial presents a self-contained account of the theory of conformal
prediction and works through several numerical examples. A more comprehensive
treatment of the topic is provided in "Algorithmic Learning in a Random World",
by Vladimir Vovk, Alex Gammerman, and Glenn Shafer (Springer, 2005).Comment: 58 pages, 9 figure
Emile Borel's difficult days in 1941
The German forces occupying Paris arrested Emile Borel and three other
members of the Acad\'emie des Sciences in October 1941 and released them about
five weeks later. Why? We examine some relevant German and French archives and
other sources and propose some hypotheses. In the process, we review how the
Occupation was structured and how it dealt with French higher education and
some French mathematicians
Two ways game-theoretic probability can improve data analysis
When testing a statistical hypothesis, is it legitimate to deliberate on the
basis of initial data about whether and how to collect further data?
Game-theoretic probability's fundamental principle for testing by betting says
yes, provided that you are testing by betting and do not risk more capital than
initially committed. Standard statistical theory uses Cournot's principle,
which does not allow such optional continuation. Cournot's principle can be
extended to allow optional continuation when testing is carried out by
multiplying likelihood ratios, but the extension lacks the simplicity and
generality of testing by betting.
Game-theoretic probability can also help us with descriptive data analysis.
To obtain a purely and honestly descriptive analysis using competing
probability distributions, we have them bet against each other using the Kelly
principle. The place of confidence intervals is then taken by a sets of
distributions that do relatively well in the competition. In the simplest
implementation, these sets coincide with R. A. Fisher's likelihood intervals
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