On the foundations of statistics:
A frequentist approach
Frank Hampel
June 1998
Abstract
A limited but basic problem in the foundations of
statistics is the following: Given a parametric model, given perhaps some
observations from the model, but given no prior information about the
parameters (``total ignorance''), what can we say about the occurrence of a
specified event A under this model in the future (prediction problem)? Or,
as probabilities are often described in terms of bets, how can we bet on A?
Bayesian solutions are internally consistent and fully conditional on
the observed data, but their ties to the observed reality and their
frequentist properties can be arbitrarily bad (unless, of course, the
assumed prior distribution happens to be the true prior). Frequentist
solutions are generally not possible with ordinary probabilities; but it is
possible to define ``successful bets'' (using upper and lower probabilities),
which even lead out of the state of total ignorance in an objective
learning process converging to the true probability model. A special
variant (successful bets on random parameter sets) provides a new and
correct interpretation of the basic idea of Fisher's ``fiducial
probabilities.''
Successful bets (which are only one-sided and not fully conditional) can
be used for inference, but not for decisions which, as in Bayesian ``fair
bets,'' require ordinary probabilities. However, it is possible to define
``best enforced fair bets'' (or corresponding probability distributions)
which solve the decision problem in a specific minimax sense. In as much as
they are also Bayes solutions, they may be called ``least unsuccessful Bayes
solutions'' (providing another candidate for ``least informative priors''). On
the other hand, among the successful bets we can select the ``least unfair
successful bets'' which in a way come closest to Bayes solutions. Several
(nontrivial) examples, mainly for two independent binomials, have already
been worked out by R. Steiner and the author.
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