Foundational issues in statistical inference C.J. Albers Groningen Biomolecular Sciences and Biotechnology Institute, University of Groningen, P.O. Box 800, NL-9700 AZ Groningen, The Netherlands. Corresponding author, c.j.albers@rug.nl O.J.W.F. Kardaun Max-Planck Institut f¨ ur Plasmaphysik, Boltzmannstraße 2, D-85748 Garching, Germany W. Schaafsma Institute of Mathematics and Computing Science, University of Groningen A.G.M. Steerneman Department of Econometrics, University of Groningen A. Stein Wageningen University, P.O. Box 16, NL-6700AA Wageningen, The Netherlands, and ITC, P.O. Box 6, NL-7500AA Enschede, The Netherlands Abstract Statistical inference is about using statistical data (x) to formulate an opinion about something that is defined well, but unknown (y). Testing a hypothesis H about y is one of the possibilities, the estimation or prediction of y is another one. We concentrate the attention on estimation or prediction in the sense that an opinion is required in the form of a probability distribution Q = Q(x) on the space Y of all theoretical possibilities. The data x being statistical, it is natural to incorporate probabilistic arguments in the context to let x speak about y. Assuming that (x, y) is the outcome of a pair (X, Y ) of random variables (in the sense of probability theory), the ‘true’ distribution P of (X, Y ) exists. It may be exactly known in simulations and in thought experiments, but it is only partially known in real-world investigations. That is why the context to let x speak about y will involve at least some specification of a family P = {P θ ; θ ∈ Θ} of theoretically possible P ’s. We assume that the probabilistic aspects of the situation are sufficiently convincing to aim at a probabilistic form of the opinion about y, given nature’s message x and ‘the context’. If a probability statement is needed about some hypothesis H with respect to y, then we construct an estimator or predictor α of the truth value of H and, if the estimator seems reasonable, we use α(x) as the (epistemic) probability of H. If a distributional inference is needed about a real-valued unknown y then, apart from using the