Abstract
The process of revising the Guide to the Expression of Uncertainty in Measurement (GUM) is ongoing. A successful revision must be theoretically sound, so it must be based on a recognized paradigm for scientific data analysis. The major candidate paradigms of statistical inference are the frequentist paradigm — where probability statements are only used to describe the frequency behaviour of processes — and the Bayesian paradigm — where probability statements are also used to describe the degree of belief of the speaker and where the subject of that belief may be the unknown value of a constant. These paradigms are incompatible, and there remains disagreement about which should form the basis of the practical procedures advocated in a revised GUM. This paper uses published results to examine foundational ideas in two different Bayesian approaches, which themselves are inconsistent at a basic, conceptual, level. So-called ‘objective’ Bayesian statistics is based on the premise that a probability distribution can accurately represent a set of information about a constant, which is a premise of the system of analysis currently favoured by the BIPM. This premise is proven to lead to a logical contradiction, which renders the theory untenable. In contrast, subjective Bayesian methods are theoretically satisfactory if they are used in their proper context, which is the context of personal decision-making. But simple examples show that the use of subjective probability in a client-centred measurement can lead to unacceptable results. These examples involve the mixing of epistemic uncertainty (the uncertainty of personal doubt) with aleatory uncertainty (the uncertainty of an unpredictable physical process). They show that these two types of uncertainty cannot always be treated as one, which is a conclusion with profound implications for the role of Bayesian statistics in science. The analysis in this paper also involves a discussion of the ‘meaning’ of probability, which has been a divisive subject for many years. It is argued here that the implied question is inappropriate: the relevant question is not “what does probability mean?” but “what role can probability meaningfully and accurately play?” The results about objective and subjective Bayesian statistics discussed in this paper support the conclusion that a sound and practical system of analysis must be based on the frequency role of probability.