Bayesian inference: an approach to statistical inference

The original Bayes used an analogy involving an invariant prior and a statistical model and argued that the resulting combination of prior with likelihood provided a probability description of an unknown parameter value in an application; the combination in particular contexts with invariance can currently be called a confidence distribution and is subject to some restrictions when used to construct confidence intervals and regions. The procedure of using a prior with likelihood has now, however, been widely generalized with invariance being extended to less restrictive criteria such as non-informative, reference, and more. Other generalizations are to allow the prior to represent various forms of background information that is available or elicited from those familiar with the statistical context; these can reasonably be called subjective priors. Still further generalizations address an anomaly where marginalization with a vector parameter gives results that contradict the term probability; these are Dawid, Stone, Zidek marginalization paradoxes; various priors for this are called targeted priors. A special case where the prior describes a random source for the parameter value is however just probability analysis but is frequently treated as a Bayes procedure. We survey the argument in support of probability characteristics and outline various generalizations of the original Bayes proposal. (C) 2010 John Wiley & Sons, Inc.