J. M. Bernardo. Modern Bayesian Inference: Foundations and Objective Methods MODERN BAYESIAN INFERENCE: FOUNDATIONS AND OBJECTIVEMETHODS

The field of statistics includes two major paradigms: frequentist and Bayesian. Bayesian methods provide a complete paradigm for both statistical inference and decision making under uncertainty. Bayesian methods may be derived from an axiomatic system and provide a coherentmethodology which makes it possible to incorporate relevant initial information, and which solvesmany of the difficulties which frequentist methods are known to face. If no prior information is to be assumed, a situation often met in scientific reporting and public decision making, a formal initial prior functionmust bemathematicallyderived from the assumedmodel. 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.