An Outline of a Unifying Statistical Theory
Frank Hampel
April 2001
Accepted paper to be presented at the Second International Symposium on
Imprecise Probabilities and their Applications. Cornell University, Ithaca, NY,
USA 26 - 29 June 2001
Abstract
A new statistical theory is outlined which builds a bridge
between frequentist and Bayesian approaches and very naturally uses
upper and lower probabilities. It started with an attempt to
investigate how far one can get with a frequentist approach; this
approach goes beyond the Neyman-Pearson and the Fisherian theory in
explicitly using intersubjective epistemic upper and lower probabilities
allowing an operational frequentist interpretation (not tied
to repetitions of an experiment), and in deriving what is valid of
Fisher's mostly misinterpreted fiducial probabilities as a very
special case within a broader framework. It formally contains the
Bayes theory as an extremal special case, but at the other extreme
it also allows starting with the state of total ignorance about the
parameter in an objective, frequentist learning process converging
to the true model, thereby solving a problem of artificial intelligence
(AI). The general theory describes (rather similar) optimal
compromises between frequentist and Bayesian approaches within
(and outside) either framework, thus also providing a new class of
"least informative priors". There is also a connection with
information theory.
Key concepts are "successful bets", more specifically "least unfair
successful bets", "cautious surprises", and "enforced fair bets",
including ``best enforced fair bets''. The main emphasis is on
prediction. When going from inference to decisions, upper and lower
probabilities (which avoid sure loss) are replaced by proper probabilities
(which are coherent), somewhat analogous to Smets' pignistic
transformation of belief functions.
Much still needs to be done, but several examples for the binomial
(the "fundamental problem of practical statistics") have been worked
out, and there are also first (rather limited) solutions for
continuous one-parameter situations, including their robustness
problem.
Keywords
Foundations of statistics, frequentist approach, Bayesian approach,
Neyman-Pearson theory, Fisherian theory; upper and lower probabilities,
aleatory and epistemic probabilities, frequentist intersubjective epistemic
probabilities, fiducial probabilities; successful bets, least unfair
successful bets, best enforced fair bets; cautious surprises; state of
total ignorance, objective learning process; inference and decisions;
prediction for binomial data.
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