A Unified Scientific Basis for Inference

Abstract

Every experiment or observational study is made in a context. This context is being explicitly considered in this book. To do so, a conceptual variable is defined as any variable which can be defined by (a group of) researchers in a given setting. Such variables are classified. Sufficiency and ancillarity are defined conditionally on the context. The conditionality principle, the sufficiency principle and the likelihood principle are generalized, and a tentative rule for when one should not condition on an ancillary is motivated by examples. The theory is illustrated by the case where a nuisance parameter is a part of the context, and for this case, model reduction is motivated. Model reduction is discussed in general from the point of view that there exists a mathematical group acting upon the parameter space. It is shown that a natural extension of this discussion also gives a conceptual basis from which essential parts of the formalism of quantum mechanics can be derived. This implies an epistemological basis for quantum theory, a kind of basis that has also been advocated by part of the quantum foundation community in recent years. Born's celebrated formula is shown to follow from a focused version of the likelihood principle together with some reasonable assumptions on rationality connected to experimental evidence. Some statistical consequences of Born's formula are sketched. The questions around Bell's inequality are approached by using the conditionality principle for each observer. The objective aspects of the world are identified with the ideal inference results upon which all observers agree (epistemological objectivity).