Bayesian Methodology in Statistics

Bayesian methods provide a complete paradigm for statistical inference under uncertainty. These may be derived from an axiomatic system and provide a coherent methodology which makes it possible to incorporate relevant initial information, and which solves many of the difficulties that frequentist methods are known to face. If no prior information is to be assumed, the more frequent situation met in scientific reporting, a formal initial prior function, the reference prior, mathematically derived from the assumed model, is used; this leads to objective Bayesian methods, objective in the precise sense that their results, like frequentist results, only depend on the assumed model and the data obtained. The Bayesian paradigm is based on an interpretation of probability as a rational conditional measure of uncertainty, which closely matches the sense of the word ‘probability’ in ordinary language. Statistical inference about a quantity of interest is described as the modification of the uncertainty about its value in the light of evidence, and Bayes’ theorem specifies how this modification should precisely be made; hence the adjective ‘Bayesian’ under which the paradigm is usually known.