Foundational issues in statistical inference

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 = {Pθ; 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 Bayesian approach, we can construct an inference by defining its distribution function Gx such that, for any z ∈ R, Gx(z) is equal to αz(x) where αz is some estimator of the truth value of Hz : y ≤ z. We claim that it is appropriate, in this respect, to use the p-value to estimate the truth value of Hz (if y is the true value of a parametric function). The imposed probabilistic coherency should not be taken too seriously because the underlying restriction of strong similarity, the Fisherian requirement that GX(Y ) follows a U(0, 1) distribution, may be ‘very’ reasonable if a distributional inference about y is required, but it is often not more than ‘fairly’ reasonable if a probability statement about Hz is needed.