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
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Seminar fuer Statistik
ETH-Zentrum, LEO D2